# A new family of inequality indices: axioms, inference and tail properties

Jonathan El Methni, Stéphane Girard, Pearl Laveur

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Jonathan El Methni, Stéphane Girard, Pearl Laveur. A new family of inequality indices: axioms, inference and tail properties. 2025. hal-05153188v2

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# A new family of inequality indices: axioms, inference and tail properties

Jonathan El Methni, Stéphane Girard\* and Pearl Laveur

Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France.

\* Corresponding author: Stephane.Girard@inria.fr

## Abstract

Inequality indices provide a quantitative framework for measuring disparity within a distribution, particularly in wealth or income. First, we introduce a unified family of inequality indices that encompasses several classical ones, including Gini, Atkinson, extended Gini, Bonferroni and Mehran indices. Second, we prove, under appropriate conditions, that indices within this family satisfy six axioms widely accepted in the literature. Third, two general estimators are proposed for this class and their asymptotic normality is established under mild assumptions. Besides, it has been observed that the Gini index is robust to changes in the highest incomes. Leveraging extreme-value theory, we prove a feature shared by the entire family: Non-discrimination of tail behaviours in terms of maximum domains of attraction. Notably, this property also holds for several alternatives to the Gini index, including those previously cited. These results are illustrated both on simulated data and on a real income data set.

**Key words:** Gini index, Atkinson index, Inequality measure, Extreme-value theory.

**MSC classification codes:** 91B82, 62G32, 62G20.

## 1 Introduction

**Framework.** According to Oxfam International, in 2023, the richest 1% of the global population own nearly half of the world's wealth<fnref n="1" /> and are responsible for 16% of global greenhouse gas emissions<fnref n="2" />. These staggering figures highlight how economic disparity is closely tied to environmental and social issues, making it essential to measure inequality in order to create fairer and more sustainable policies.

The idea of measuring inequality began to take shape in 1912, when the Italian economist Corrado Gini introduced his eponymous index [25], based on the Lorenz curve [37]. This index is used to account for the dispersion of a variable within a population, making it possible to quantify the inequality level in a wealth distribution. Many alternative measures have been introduced such as Atkinson [2], extended Gini [52], Bonferroni [7], Mehran [39], Theil [51] indices or the coefficient of variation [41, Section 5.c.(i)].

More generally, an inequality index can be defined as a function that associates a non negative number with a distribution and whose lower bound means perfect equality (everyone has the same share of wealth) and upper one represents perfect inequality (one person has all the wealth). In economics, the measurement of income/wealth inequalities should additionally follow a few basic axioms: Normalisation, scale-invariance, aggregativity, transfer, anonymity and population principles, see Appendix A for details.

<footnote n="1">¹https://www.oxfam.org.tw/en/news-and-publication/davos-2023</footnote>

<footnote n="2">²https://www.oxfam.org/en/press-releases/richest-1-emit-much-planet-heating-pollution-two-thirds-humanity</footnote>
Given the large number of inequality indices, the question of unifying them within families has arisen in the literature. For example, [15, 39, 54] define families of indices based on the Lorenz curve, regardless of the set of axioms they satisfy. More specifically, the class in [15] encompasses Gini, Bonferroni and Mehran indices, but notably excludes the Atkinson index. The families introduced in [39, 54] include Gini and Mehran indices, while excluding, in particular, Atkinson and Bonferroni indices. An alternative point of view is to pick an axiom and select compatible indices. This is the approach adopted in [47] which relies in particular on additive decomposability, an axiom stronger than aggregativity, which is not fulfilled by the Gini index for instance. The resulting class comprises the Theil index, but excludes all other indices mentioned above.

Another line of works has recently arisen, focused on the upper tail properties of inequality indices. The Gini index, in particular, does not reflect inequality among the very wealthy. This robustness with respect to top tail changes may lead to underestimate actual inequality, despite the fact that increases in top income shares can significantly affect overall inequality [3, 4, 19]. This feature is pointed out in [33, Proposition 1], where a Pareto distribution with shape parameter $\gamma = 2/3$ (see Table 1 in C for its parameterisation) yields the same Gini index (1/2) as any exponential distribution. Since these two distributions belong to different maximum domains of attraction (see Subsection 3.1 for a short introduction to extreme-value theory), this implies that the Gini index does not distinguish between fundamentally different tail behaviours. Interestingly, this kind of robustness with respect to tail heaviness has also recently been examined for scoring rules [9].

**Our contributions.** We first define a new family of indices that unifies popular inequality measures such as Gini, Atkinson, extended Gini, Bonferroni and Mehran indices, which, to the best of our knowledge, have not been previously considered all together in the literature. This unification makes it possible to perform calculations and to derive theoretical properties at the family level. As a prominent example, we prove that all the family members fulfil the basic axioms. This also allows us to propose two general non-parametric estimators (dedicated respectively to individual data and grouped data) i.e. valid for the whole family of indices. Their asymptotic normality is established under mild conditions.

Our second contribution is to rigorously investigate the non-discrimination property of maximum domains of attraction shared by this whole family of inequality indices. The seminal result in [33] (established on a particular example) is extended in several directions. Leveraging extreme-value theory, we prove that the non-discrimination property holds: First, for the proposed family of inequality indices, and second, between and within maximum domains of attraction. Even though the study of variability measures through the lens of extreme-value theory has recently emerged [21, 22], the application to inequality indices remains largely unexplored. Apart from [5] dedicated to estimating the Gini index for heavy-tailed distributions, we are not aware of any other such work.

**Outline of the paper.** This paper is organised as follows. Section 2 defines the new family of inequality indices, establishes its relationship with various known indices, and verifies the selected axioms (listed in Appendix A) on this class. Section 3 provides basic tools from extreme-value theory used to study the non-discrimination property of upper tail behaviours. Section 4 discusses the estimation of indices in our family (both for individual and grouped data), including asymptotic normality results. Finally, Sections 5 and 6 illustrate these theoretical findings on simulations and on a real income data set, respectively. Proofs are postponed to Appendix B.
## 2 A new set of inequality indices

The presentation of the new family of inequality indices in Subsection 2.1 is preceded by the definition of key mathematical tools and a discussion of their basic properties. Then, connections with well-known indices such as the Gini index are highlighted in Subsection 2.2, where a new subclass of indices is also proposed. Finally, Subsection 2.3 states that the considered indices satisfy the key axioms (A₁)–(A₆) presented in Appendix A.

### 2.1 Definitions and first properties

Our basic assumption is:

(H₁) The random variable of interest X is continuous, and a.s. positive. Its cumulative distribution function (c.d.f.) F is strictly increasing on its support, and its expectation $\mu = E(X)$ exists with $\mu \in (0, +\infty)$.

Under assumption (H₁), the survival function associated with X is denoted by $\bar{F} := 1 - F$ and the associated density by f. The quantile function q is defined by $q(u) = \bar{F}^{-1}(u)$ for all $u \in (0, 1)$. The tail quantile function U is defined by $U(t) = q(1/t)$ for all $t \ge 1$. In particular, the starting point is $x_1 = U(1)$ and the endpoint is given by $x_\infty = U(\infty)$.

The proposed family of inequality indices is based on a weight function $g: [0, 1) \to [0, +\infty)$. Assumption (H₂) imposes shape constraints on g such that the associated inequality index satisfy the basic axioms (A₁)–(A₆), see Subsection 2.3 and Appendix A.

(H₂) The density g is defined on $[0, 1)$, continuously differentiable on $(0, 1)$ and increasing. Its associated c.d.f. is denoted by G.

For all density g satisfying (H₂), consider the set

$$ \mathcal{K}(g) = \begin{cases} (-\infty, 0) \cup (0, 1] & \text{if } g \text{ is strictly increasing and } g(0) = 0, \\ (-\infty, 0) \cup (0, 1) & \text{otherwise.} \end{cases} $$

As an example, the density of the Beta($\alpha, \beta$) distribution

$$ u \in [0, 1) \mapsto g_{\alpha, \beta}(u) = \frac{1}{B(\alpha, \beta)} u^{\alpha-1} (1-u)^{\beta-1}, \qquad (2.1) $$

where B is the Beta function, satisfies (H₂) with $(\alpha, \beta) \in \mathcal{B} = [1, +\infty) \times (0, 1]$ and

$$ \mathcal{K}(g_{\alpha, \beta}) = \begin{cases} (-\infty, 0) \cup (0, 1] & \text{if } \alpha > 1, \\ (-\infty, 0) \cup (0, 1) & \text{if } \alpha = 1, \end{cases} $$

see Lemma 1 in Appendix B.1. Clearly, (H₂) is preserved by mixing. Indeed, if $g^{[0]}$ and $g^{[1]}$ satisfy (H₂), then, for all $p \in [0, 1]$, the mixture density $g^{[p]} = pg^{[1]} + (1-p)g^{[0]}$ also satisfies (H₂).

Under (H₁) and (H₂), the main building block of the new set of inequality indices is

$$ D_{\kappa,g}(X) = \int_{0}^{1} q(u)^{\kappa} g(u) du = E[X^{\kappa} g(\bar{F}(X))], $$

where $\kappa \in \mathcal{K}(g)$. The existence of $D_{\kappa,g}(X)$ is established in Lemma 4 in Appendix B.1. It can be interpreted as a probability weighted moment of X, extending the definition in [31] to the case where g is not necessarily a power function, and the definition in [13] to $\kappa \neq 1$. Note that $D_{1,g}(X)$ is a so-called signed Choquet integral with weighting function $g(1-\cdot)$, see [21, Subsection 2.2], and is also a monotonic risk measure, see [21, Lemma 4.2(a)]. We are now in position to define the population version of the new inequality index which depends on a real parameter $\kappa$ and a weight function g.
**Definition 1** (New family $(I_{\kappa,g})$ of inequality indices). Suppose $(H_1)$ and $(H_2)$ hold. Then, for all $\kappa \in K(g)$, the following index is introduced:

$$I_{\kappa,g}(X) = 1 - \frac{D_{\kappa,g}^{1/\kappa}(X)}{D_{1,1}(X)} = 1 - \frac{D_{\kappa,g}^{1/\kappa}(X)}{\mu}.$$

We refer to Subsection 2.2 for examples of $\kappa$ and $g$ providing existing as well as new inequality indices. It is also proved in Subsection 2.3 that Definition 1 yields valid inequality indices verifying the six basic axioms (A1)-(A6), see Appendix A. To this end, let us first introduce the empirical counterpart of the index $I_{\kappa,g}$. Let $\mathcal{X}_n = \{X_1, \dots, X_n\}$ be a sample of independent random variables with same distribution as $X$. Let $X_{1,n} \le \dots \le X_{n,n}$ be the associated order statistics. Then, $D_{\kappa,g}(X)$ can be estimated by approximating the quantile $q(u)$ by $X_{n-i+1,n}$ for all $u \in [(i-1)/n, i/n)$, leading to

$$\widehat{D}_{\kappa,g}(\mathcal{X}_n) = \sum_{i=1}^{n} X_{n-i+1,n}^{\kappa} (G(i/n) - G((i-1)/n)). \quad (2.2)$$

The sample version of $I_{\kappa,g}(X)$ is thus defined by

$$\widehat{I}_{\kappa,g}(\mathcal{X}_n) = 1 - \frac{\widehat{D}_{\kappa,g}^{1/\kappa}(\mathcal{X}_n)}{\hat{\mu}_1}, \quad (2.3)$$

with for $p \in \mathbb{R}$:

$$\hat{\mu}_p = \frac{1}{n} \sum_{i=1}^{n} X_i^p, \quad (2.4)$$

the empirical counterpart of $\mu_p = \mathbb{E}(X^p)$ (whenever it exists) so that $\mu_1 = \mu$.

## 2.2 Classical and novel inequality indices

The set $(I_{\kappa,g})$ includes several well-known indices presented hereafter, see also the first three columns of Table 2 in Appendix C for a short summary. Throughout, $X$ denotes a random variable satisfying (H1).

**Gini index.** It relies on the Lorenz curve, a graphical tool to represent the income distribution in a population. The Lorenz curve is obtained by plotting the Lorenz function $p \in [0, 1] \mapsto L(p)$ giving at each point $p$ the cumulative proportion of income held by the poorest $100p\%$ of the population:

$$L(p) = \frac{1}{\mu} \int_{0}^{p} F^{-1}(u) du = \frac{1}{\mu} \int_{1-p}^{1} q(u) du.$$

We refer to [37] for the early definition of the empirical counterpart of $L$ and to [23] for the above population version. The Gini index [25], denoted by $I_{\text{Gini}}$, is defined as $1 - 2S$, where $S$ is the area under the Lorenz curve:

$$I_{\text{Gini}}(X) = 1 - 2 \int_{0}^{1} L(p) dp = 1 - \frac{2}{\mu} \int_{0}^{1} q(u) u du. \quad (2.5)$$

Clearly, $I_{\text{Gini}}$ is a particular case of $I_{\kappa,g}$ obtained with $\kappa = 1$ and $u \in [0, 1] \mapsto g_{2,1}(u) = 2u$, the density of the Beta(2, 1) distribution, which satisfies (H2) as shown in Lemma 1 (see Appendix B.1).

**Atkinson index.** The latter relies on the $(1-\eta)$th moment of $X$, with $\eta \in (0, +\infty) \setminus \{1\}$. The Atkinson index [2] is defined as

$$I_{\text{Atkinson},\eta}(X) = 1 - \frac{\mu_{1-\eta}^{1/(1-\eta)}}{\mu} = 1 - \frac{1}{\mu} \left( \int_{0}^{1} q(u)^{1-\eta} du \right)^{1/(1-\eta)}. \quad (2.6)$$
This index $I_{Atkinson,\eta}$ is clearly a particular case of $I_{\kappa,g}$, with $\kappa = 1 - \eta \in \mathcal{K}(1)$ and $u \in [0, 1] \mapsto g_{1,1}(u) = 1$, the density of the Beta(1, 1) distribution which satisfies (H$_2$), see Lemma 1. The parameter $\eta$ is the degree of inequality-aversion or, in the case of incomes, the relative sensitivity to transfers at different income levels. As $\eta$ increases, more weight is associated with transfers among the poorest individuals and less weight to transfers among the richest.

**The class of extended Gini indices.** It is also based on the Lorenz curve with an additional parameter $\nu \ge 1$. The extended Gini index [52] is defined as

$$I_{EG,\nu}(X) = 1 - \nu(\nu - 1) \int_{0}^{1} (1-p)^{\nu-2} L(p) dp = 1 - \frac{\nu}{\mu} \int_{0}^{1} q(u) u^{\nu-1} du. \quad (2.7)$$

The extended Gini index $I_{EG,\nu}$ is thus a particular case of $I_{\kappa,g}$ with $\kappa = 1$ and $u \in [0, 1] \mapsto g_{\nu,1}(u) = \nu u^{\nu-1}$, the density of the Beta($\nu$, 1) distribution satisfying (H$_2$) when $\nu \ge 1$, see Lemma 1. In addition, the tuning parameter $\nu$ controls the sensitivity of $I_{EG,\nu}$ to different parts of the distribution, as detailed below in the case of incomes [34]:

* The case $\nu = 1$ can be discarded since $I_{EG,1}(X) = 0$ for all integrable random variable $X$.

* If $\nu \in (1, 2)$, a given difference in income among the richest individuals counts more towards overall inequality than the same difference among the poorest individuals.

* If $\nu = 2$, no weights are assigned to Lorenz curve points and thus $I_{EG,2}(X) = I_{Gini}(X)$.

* If $\nu > 2$, a given difference in income among the poorest individuals counts more towards overall inequality than the same difference among the richest individuals.

* Finally, the extreme case $\nu \to +\infty$ yields $\nu \to 1 - x_1/\mu$, such that only the lowest income counts, see [52, p.618].

**The class of Bonferroni indices.** This family of indices is introduced in [32], using the Lorenz curve. These indices depend on a parameter $h \in \mathbb{N} \setminus \{0\}$ and they generalise the Bonferroni index (obtained when $h = 1$), see [7]. One way to define this class of indices is:

$$I_{\text{Bonferroni},h}(X) = 1 - h \int_{0}^{1} \frac{(1-p)^{h-1}}{p} L(p) dp = 1 - \frac{h}{\mu} \int_{0}^{1} q(u) \left( \int_{1-u}^{1} \frac{(1-p)^{h-1}}{p} dp \right) du. \quad (2.8)$$

It is a particular case of $I_{\kappa,g}$ with $\kappa = 1$ and $g(u) = h \int_{1-u}^{1} (1-p)^{h-1}/p dp$, for which (H$_2$) holds for all $h \in [1, +\infty)$, see Lemma 5 in Appendix B.1. This remark provides a continuous extension of the class of Bonferroni indices. Note that the Bonferroni index ($h = 1$) is associated with $g(u) = -\log(1 - u)$. Indices $I_{\text{Bonferroni},h}$ give greater importance to inequality in lower incomes. Furthermore, as $h$ increases, this difference becomes increasingly marked.

**The class of Mehran inequality indices.** This family of indices is defined in [39], using an averaging of the deviations of $p \in (0, 1) \mapsto F^{-1}(p)$ from the expectation $\mu$. Indices of this class depend on a score function $W$ i.e. a continuous function defined on $(0, 1]$ and independent of $F$, such that $\int_{0}^{1} W(p) dp = 0$ and $W(1) = 1$. These indices are defined as

$$I_{W}(X) = \frac{1}{\mu} \int_{0}^{1} [F^{-1}(p) - \mu] W(p) dp = 1 - \frac{1}{\mu} \int_{0}^{1} q(u) [1 - W(1 - u)] du.$$

If $W$ is strictly increasing and such that (H$_2$) holds for $g(u) = 1 - W(1 - u)$, then $I_W$ is the particular case of $I_{\kappa,g}$, with $\kappa = 1$. The score function $W$ allows for control over which parts of the distribution
have the greatest impact on the inequality index. For instance, in the case of incomes, if W is concave, it emphasises disparities among lower incomes, while a convex W puts more weight on disparities at higher income levels. These indices include extended Gini indices with $W_{\nu,1}(u) = 1 - \nu(1 - u)^{\nu-1}$ (associated with the Beta density $g_{\nu,1}$, see (2.1)) and therefore also the Gini index with $W_{2,1}(u) = 2u - 1$. This family also encompasses the class of Bonferroni indices with $W(u) = 1 - g(1 - u)$ and $g(u) = h \int_{1-u}^{1} (1 - p)^{h-1}/p \, dp$, for all $h \ge 1$. The special case of $W(p) = 1 - 3(1 - p)^2$ leads to the well-known Mehran index denoted by $I_{\text{Mehran}}$, with $u \in [0, 1] \mapsto g_{3,1}(u) = 3u^2$, the density of the Beta(3, 1) distribution satisfying (H₂), see Lemma 1. Note that it coincides with the extended Gini index with parameter $\nu = 3$ and can be written as:

$$I_{\text{Mehran}}(X) = I_{\text{EG},3}(X) = 6 \int_{0}^{1} (p - L(p))(1 - p)dp. \qquad (2.9)$$

Finally, a new subclass of inequality indices based on the Beta density is introduced below.

**The class of Beta indices.** For all $(\alpha, \beta) \in \mathcal{B}$ and $\kappa \in \mathcal{K}(g_{\alpha,\beta})$, Beta indices are denoted by $I_{\text{Beta},(\alpha,\beta,\kappa)}$ and defined as:

$$I_{\text{Beta},(\alpha,\beta,\kappa)}(X) = I_{\kappa,g_{\alpha,\beta}}(X) = 1 - \frac{\left(\int_{0}^{1} q(u)^{\kappa} u^{\alpha-1} (1-u)^{\beta-1} du\right)^{1/\kappa}}{\mu B(\alpha, \beta)^{1/\kappa}}. \qquad (2.10)$$

Recall that (H₂) holds for $g_{\alpha,\beta}$, as shown in Lemma 1. Sensitivity to different parts of the distribution is controlled both by the parameter $\kappa$ as for the Atkinson index, and by the pair $(\alpha, \beta)$ parameterising the weight function $g_{\alpha,\beta}$, as in extended Gini indices. This new subclass includes $I_{\text{EG},\nu}$ with $\kappa = 1$ and $(\alpha, \beta) = (\nu, 1)$, especially the Gini index with $\alpha = 2$ and the Mehran index with $\alpha = 3$. The index $I_{\text{Atkinson},\eta}$ is also included with $\kappa = 1 - \eta$ and $(\alpha, \beta) = (0, 0)$.

## 2.3 Verification of six axioms

Let us show that the family of inequality indices fulfils the six axioms (A₁)-(A₆) listed in Appendix A (at least in a weak sense). This provides a unified proof of the axiomatic properties for Gini, Atkinson, extended Gini, Bonferroni and Mehran indices. In contrast, the Theil index and the coefficient of variation do not belong to the proposed family and do not satisfy the normalisation axiom (A₁), see [10, Table 3.5]. The first proposition addresses the verification of Axioms (A₁) and (A₂) for the population version of the inequality index $I_{\kappa,g}$.

**Proposition 1** (Normalisation and scale-invariance). Suppose (H₁) and (H₂) hold. Then, $I_{\kappa,g}$ satisfies (A₁) and (A₂) for all $\kappa \in \mathcal{K}(g)$.

The following second proposition deals with Axioms (A₄), (A₅) and (A₆) on the sample version of the inequality index.

**Proposition 2** (Tranfer, anonymity and population principles). Suppose (H₁) and (H₂) hold. Then, $\widehat{I}_{\kappa,g}$ satisfies (A₄), (A₅) and (A₆) for all $\kappa \in \mathcal{K}(g)$.

Only few inequality indices satisfy the aggregativity/decomposability principle (A₃), and even fewer when restricted to those satisfying all six axioms, see [8, Table I] and [10, Table 3.5]. One can for instance cite the Atkinson index in the ($I_{\kappa,g}$) family, see Subsection 2.2, and the index introduced in [33]. In the result below, the decomposability of the index $I_{\kappa,g}$ is investigated when applied to a mixture of distributions with non overlapping supports. Note that the components of the mixture play the role of subgroups in the sample version [8, Definition 1]. In addition, two sufficient conditions on g are provided so that the index $I_{\kappa,g}$ associated with this mixture is aggregative according to disjoint subgroups.
**Proposition 3** (Aggregativity/decomposability principle). Let $m \in \mathbb{N} \setminus \{0\}$. For all $j \in \{1, \dots, m\}$, let $X^{[j]}$ be a random variable satisfying (H₁) with c.d.f. $F^{[j]}$ and expectation $\mu^{[j]}$. Suppose the m associated supports are non overlapping. For all $j \in \{1, \dots, m\}$, let $p^{[j]} \in (0, 1)$ such that $\sum_{j=1}^{m} p^{[j]} = 1$. Let X be a random variable with associated c.d.f. $F = \sum_{j=1}^{m} p^{[j]} F^{[j]}$ and denote by $\mu$ its expectation. Then,

(i) X satisfies (H₁). Moreover, for all g satisfying (H₂) and all $\kappa \in K(g)$,

$$I_{\kappa,g}(X) = 1 - \frac{1}{\mu} \left( \sum_{j=1}^{m} \omega^{[j]} \left\{ \mu^{[j]} \left[ 1 - I_{\kappa,g^{[j]}}(X^{[j]}) \right] \right\}^{\kappa} \right)^{1/\kappa},$$

where for all $j \in \{1, \dots, m\}$,

$$\omega^{[j]} = G\left(\sum_{i=j}^{m} p^{[i]}\right) - G\left(\sum_{i=j+1}^{m} p^{[i]}\right) \quad \text{and} \quad g^{[j]}(u) = \frac{p^{[j]}}{\omega^{[j]}} g\left(p^{[j]}u + \sum_{i=j+1}^{m} p^{[i]}\right), \quad u \in [0, 1).$$

(ii) The only solution of $g = g^{[j]}$ for all $j \in \{1, \dots, m\}$ and all $(p^{[1]}, \dots, p^{[m]}) \in (0, 1)^m$ satisfying (H₂) is $g_{1,1} \equiv 1$, the density of the Beta(1, 1) distribution. Then,

$$I_{\kappa,1}(X) = 1 - \frac{1}{\mu} \left\{ \sum_{j=1}^{m} p^{[j]} \left[ \mu^{[j]} (1 - I_{\kappa,1}(X^{[j]})) \right]^{\kappa} \right\}^{1/\kappa}$$

satisfies (A₃).

(iii) Suppose g satisfies (H₂) and is linear. Then, $g = g_{2,1}$ i.e. $g(u) = 2u$, for all $u \in [0, 1]$. Moreover,

$$I_{\kappa,g_{2,1}}(X) = 1 - \frac{1}{\mu} \left\{ \sum_{j=1}^{m} \left[ (p^{[j]})^2 (\mu^{[j]}(1 - I_{\kappa,g_{2,1}}(X^{[j]})))^{\kappa} + 2\mathbb{E}((X^{[j]})^{\kappa}) p^{[j]} \sum_{i=j+1}^{m} p^{[i]} \right] \right\}^{1/\kappa}$$

satisfies (A₃) for $\kappa = 1$.

Proposition 3(i) establishes, for the entire family $(I_{\kappa,g})$, a weak form of aggregativity, in the sense that each group-specific index is associated with its own weight function. Proposition 3(ii) shows that the aggregativity principle holds for the Atkinson index, in the case of disjoint supports. Proposition 3(iii), in turn, provides another weak form of aggregativity for indices $I_{\kappa,g_{2,1}}$ in which the $\kappa$-th moment of each group distribution is involved. It also confirms that the Gini index satisfies the aggregativity principle in the case of disjoint supports, see [49] for a proof in the sample version case.

We refer to Subsection 4.2 for an application to the estimation in the case of grouped frequency distributions.

# 3 Analysing tail properties of inequality indices

Necessary preliminaries from extreme-value theory on tail behaviours are presented in Subsection 3.1. Subsection 3.2 then provides some computations of inequality indices for three illustrative distributions, each having a different tail behaviour. Finally, Subsection 3.3 states two properties of non-discrimination with respect to upper tails, shared by all indices in $(I_{\kappa,g})$.
## 3.1 Extreme-value theory

The extreme-value theorem [17, 27] establishes the limit distribution of the maximum of an independent sample from a c.d.f. $F$. It depends on a real parameter $\gamma$, called the extreme-value index of the parent distribution. The first-order condition (C₁) below is necessary and sufficient for the distribution of the sample maximum to converge to the so-called extreme-value distribution, see [12, Theorem 1.1.2]. The random variable $X$, or its associated c.d.f., is said to satisfy the first-order condition if the following property holds:

(C₁) There exist a positive (auxiliary) function $a$ and $\gamma \in \mathbb{R}$ such that for all $t > 0$:

$$ \lim_{x \to +\infty} \frac{U(tx) - U(x)}{a(x)} = k_{\gamma}(t), $$

where

$$ k_{\gamma}(t) = \int_{1}^{t} u^{\gamma-1} du = \begin{cases} \frac{t^{\gamma-1}}{\gamma} & \text{if } \gamma \neq 0, \\ \log(t) & \text{if } \gamma = 0. \end{cases} \qquad (3.1) $$

According to the position of $\gamma$ with respect to zero, three subclasses of distributions are defined, called maximum domains of attraction (MDAs), corresponding to different tail behaviours:

* $F \in \text{MDA}(\text{Fréchet})$ if $\gamma > 0$. This coincides with the set of heavy-tailed distributions whose survival function decreases as a power function, see [16, Section 3.3.1].

* $F \in \text{MDA}(\text{Weibull})$ if $\gamma < 0$, including short-tailed distributions whose endpoint is finite, see [16, Section 3.3.2].

* $F \in \text{MDA}(\text{Gumbel})$ if $\gamma = 0$, including light-tailed distributions whose survival function decreases at an exponential rate, see [16, Section 3.3.3].

Some examples of usual distributions belonging to a MDA are given in [16, Tables 3.4.2–3.4.4].

The characterisation of MDAs is based on the theory of regularly-varying functions [6]. Recall that a strictly positive measurable function $\varphi$ is regularly-varying at infinity with index $\rho \in \mathbb{R}$ if, for all $t > 0$,

$$ \lim_{x \to +\infty} \frac{\varphi(tx)}{\varphi(x)} = t^{\rho}. $$

This property is denoted by $\varphi \in \mathcal{RV}_{\rho}$. In the case where $\rho = 0$, $\varphi$ is said to be slowly-varying. Clearly, $\varphi \in \mathcal{RV}_{\rho}$ if and only if $\varphi(x) = x^{\rho}L(x)$ with $L \in \mathcal{RV}_{0}$.

As an example of the interplay between extreme-value theory and regularly-varying functions, one has the nice characterisation [12, Theorem 1.2.1]: $F \in \text{MDA}(\text{Fréchet})$ if and only if $\bar{F} \in \mathcal{RV}_{-1/\gamma}$ with $\gamma > 0$. This characterisation helps establish several classical results and is used for instance in the proof of Lemma 9, see Appendix B.1.

## 3.2 Examples

Let us provide some examples of calculation of Beta indices (2.10) for three distributions: Pareto in MDA(Fréchet), Beta in MDA(Weibull) and Weibull in MDA(Gumbel), see Table 1 in Appendix C for their parameterisation.
**Example 1** (Calculation of $I_{\text{Beta},(\alpha,\beta,\kappa)}$ in the three maximum domains of attraction). Let $(\alpha, \beta) \in \mathcal{B}$ and $\kappa \in \mathcal{K}(g_{\alpha,\beta})$.

(i) If $X_\gamma \sim \text{Pareto}(1, \gamma)$ with $\gamma \in (0, 1)$, then $X_\gamma$ satisfies (H$_1$) and

$$I_{\text{Beta},(\alpha,\beta,\kappa)}(X_\gamma) = 1 - (1 - \gamma) \left( \frac{B(\alpha - \kappa\gamma, \beta)}{B(\alpha, \beta)} \right)^{1/\kappa}.$$

(ii) If $Y_\theta \sim \text{Beta}(\theta, 1)$ with $\theta > 0$, then $Y_\theta$ satisfies (H$_1$) and

$$I_{\text{Beta},(\alpha,\beta,\kappa)}(Y_\theta) = \begin{cases} 1 - \left(1 + \frac{1}{\theta}\right) \left(\frac{B(\alpha, \kappa/\theta + \beta)}{B(\alpha, \beta)}\right)^{1/\kappa} & \text{if } \kappa/\theta + \beta > 0, \\ 1 & \text{otherwise.} \end{cases}$$

(iii) If $Z_b \sim \text{Weibull}(1, b)$ with $b > 0$, then $Z_b$ satisfies (H$_1$) and

$$I_{\text{Beta},(\alpha,\beta,\kappa)}(Z_b) = \begin{cases} 1 - \frac{b}{\Gamma(1/b)B(\alpha,\beta)^{1/\kappa}} \left(\int_0^1 (\log(1/u))^{\kappa/b} u^{\alpha-1} (1-u)^{\beta-1} du\right)^{1/\kappa} & \text{if } \kappa/b + \beta > 0, \\ 1 & \text{otherwise,} \end{cases}$$

where $\Gamma$ denotes the Gamma function.

The application to standard inequality indices listed in Subsection 2.2 (Gini, Atkinson, extended Gini and Mehran indices) is then straightforward using appropriate choices of $(\alpha, \beta, \kappa)$ and are given in Table 2 of Appendix C. This table summarises several results scattered throughout the literature and calculated on a case-by-case basis: The Gini index for Pareto, Beta, Weibull and exponential distributions [26, 33], the extended Gini index for Pareto and exponential distributions [54] and the Bonferroni index for Pareto, Beta, Weibull, exponential and uniform distributions [26, 43, 50].

Let us remark that, in the case (i) of the Pareto(1, $\gamma$) distribution, $I_{\text{Beta},(\alpha,\beta,\kappa)}(X_\gamma) \to 0$ as $\gamma \to 0^+$ while $I_{\text{Beta},(\alpha,\beta,\kappa)}(X_\gamma) \to 1$ as $\gamma \to 1^-$. One can then suspect that any value of inequality index in $(0, 1)$ can be reached in the MDA(Fr\'echet). We shall show in the next section that it is indeed the case in any maximum domain of attraction.

## 3.3 Non-discrimination of upper tail behaviours

As pointed out in [3, Section 2], the Gini index is robust with respect to changes among top values, especially for incomes or wealth distributions.

More specifically, a heavy-tailed Pareto distribution can have the same Gini index as a light-tailed exponential distribution, as highlighted in [33, Proposition 1] thanks to an adapted choice of the distribution parameter. Note that this result can be recovered as a particular case of Example 1(i,iii), see the second line of Table 2 in Appendix C, taking $\gamma = 2/3$ and $b = 1$. It even appears that the Gini index associated with a Pareto distribution is $\gamma/(2 - \gamma)$ which is a monotonic function of the shape parameter $\gamma \in (0, 1)$. This suggests that a c.d.f. in MDA(Fr\'echet) may reach any Gini index in the unit interval by adjusting a shape parameter. Consequently, the Gini index is unlikely to discriminate between maximum domains of attraction.

To establish such a result on the entire family $(I_{\kappa,g})$, the next two assumptions are introduced. They control the behaviour of the weight function $g$ on the boundaries of the unit interval.

(H$_3$) There exists $\alpha \ge 1$ such that $g(u) = O(u^{\alpha-1})$ as $u \to 0$.

(H$_4$) There exist $L \in \mathcal{RV}_0$ and $\beta \in (0, 1]$ such that, for all $u \in [0, 1)$,

$$g(u) = (1 - u)^{\beta-1} L \left( \frac{1}{1 - u} \right) \quad \text{and} \quad \int^{+\infty} \frac{L(t)}{t} dt = +\infty.$$
Note that (H3) and (H4) are preserved by mixing, see Lemma 2 in Appendix B.1. Under (H3), necessarily $g(0) < \infty$ which is coherent with the increasingness of $g$ in (H2), while under (H4), one may have $g(1) = +\infty$ if $\beta < 1$ or $g(1) < \infty$ when $\beta = 1$ and $L$ is constant. Moreover, under (H2) and (H4), one can assume that $L$ is normalised without loss of generality, and therefore $xL'(x)/L(x) \to 0$ as $x \to +\infty$, see [6, Page 15]. This in turn implies that

$$g'(u) \sim (1 - \beta)(1 - u)^{\beta-2} L\left(\frac{1}{1 - u}\right), \text{ as } u \to 1^{-}. \quad (3.2)$$

The Beta density $g_{\alpha, \beta}$ defined in (2.1) satisfies (H3) and (H4) for all $(\alpha, \beta) \in \mathcal{B}$, see Lemma 1 in Appendix B.1. Consequently, the weight functions associated with Gini, Atkinson, extended Gini and Mehran indices also fulfil (H3) and (H4). This is still the case for weight functions associated with the Bonferroni class, see Lemma 5 in Appendix B.1.

We rigorously state below that all inequality indices in $(I_{\kappa,g})$ do not discriminate between maximum domains of attraction, see Theorem 1: Any value in the open unit interval can be obtained in any maximum domain of attraction. The proof relies on the three distributions exhibited in Example 1.

**Theorem 1** (Non-discrimination between maximum domains of attraction). Let $g$ satisfy (H2) and $\kappa \in \mathcal{K}(g)$. Let $I \in (0, 1)$.

(i) Suppose (H3) holds and $(\kappa, \alpha) \neq (1, 1)$. Then, there exists $X \in \text{MDA}(\text{Fréchet})$ such that $I_{\kappa,g}(X) = I$.

(ii) Suppose (H4) holds. Then, there exists $Y \in \text{MDA}(\text{Weibull})$ such that $I_{\kappa,g}(Y) = I$.

(iii) Suppose (H3) holds and $(\kappa, \alpha) \neq (1, 1)$. Then, there exists $Z \in \text{MDA}(\text{Gumbel})$ such that $I_{\kappa,g}(Z) = I$.

It is possible to go further and establish that any pair of extreme-value index (less that one) and inequality index (in the open unit interval) can also be reached.

**Theorem 2** (Non-discrimination within maximum domains of attraction). Let $g$ satisfy (H2)-(H4) and $\kappa \in \mathcal{K}(g)$ with $(\kappa, \alpha) \neq (1, 1)$. Then, for all $(\gamma, I) \in (-\infty, 1) \times (0, 1)$, there exists a random variable $X$ with extreme-value index $\gamma$ such that $I_{\kappa,g}(X) = I$.

As a consequence, the considered inequality indices and the tail behaviour appear to be independent features of a distribution.

# 4 Inference

First, the asymptotic normality of $\hat{I}_{\kappa,g}$ dedicated to individual data is established in Subsection 4.1. We also provide sufficient conditions for this property to hold for three illustrative distributions. Subsection 4.2 introduces another estimator of $\hat{I}_{\kappa,g}$, tailored to situations where data are only available in grouped form. Its asymptotic normality is also discussed.

## 4.1 Asymptotic normality, individual data case

Here, the focus is on two subgroups of $(I_{\kappa,g})$ which encompass indices considered in Subsection 2.2: $g \equiv 1$ or $\kappa = 1$. In these two cases, the following theorem provides the asymptotic normality of the estimator $\hat{I}_{\kappa,g}$ defined in (2.3).

**Theorem 3** (Asymptotic normality of $\hat{I}_{\kappa,g}$). Suppose (H1) and (H2) hold and $\mu_2$ is finite.
(i) Let $g \equiv 1$ and $\kappa \in (-\infty, 0) \cup (0, 1)$ such that $\mu_{2\kappa}$ is finite if $\kappa < 0$. Then,

$$ \sqrt{n} \left( \frac{\widehat{I}_{\kappa,1}(\mathcal{X}_n) - I_{\kappa,1}(X)}{1 - I_{\kappa,1}(X)} \right) \xrightarrow[n \to +\infty]{d} \mathcal{N}(0, \sigma_{\kappa}^2(F)), \quad (4.1) $$

where

$$ \sigma_{\kappa}^2(F) = \left(\frac{\mu_2}{\mu^2} - 1\right) - \frac{2}{\kappa} \left(\frac{\mu_{\kappa+1}}{\mu_{\kappa}} - 1\right) + \frac{1}{\kappa^2} \left(\frac{\mu_{2\kappa}}{\mu_{\kappa}^2} - 1\right). \quad (4.2) $$

(ii) Suppose $g$ is strictly increasing with $g(0) = 0$ and $(\mathbf{H}_4)$ holds with $\beta \in (1/2, 1]$. Moreover, suppose;

(1) There exist $s \ge 2$ and $\delta \in (0, \beta/2 - 1/4)$ such that $g'(u) = \mathcal{O}(u^{-3/2+1/s+\delta})$ as $u \to 0^+$.

(2) Condition (C1) holds with $\gamma < \min(\beta - 1/2, 1/s)$.

Then,

$$ \sqrt{n} \left( \frac{\widehat{I}_{1,g}(\mathcal{X}_n) - I_{1,g}(X)}{1 - I_{1,g}(X)} \right) \xrightarrow[n \to +\infty]{d} \mathcal{N}(0, \sigma^2(J, F)), $$

where

$$ J(u) = \frac{1}{\mu} - \frac{g(1-u)}{D_{1,g}(X)}, \quad u \in (0, 1], $$

$$ \sigma^2(J, F) = \int_0^{+\infty} \int_0^{+\infty} J(F(x))J(F(y))[F(\min(x, y) - F(x)F(y))]dxdy. \quad (4.3) $$

Theorem 3 establishes two new asymptotic results: (i) for the empirical Atkinson index and (ii) for the empirical Bonferroni index ($h = 1$). It also allows us to find back in a unified way some known asymptotic normality behaviours, see [30, Section 9(b)] for the Gini index and [54, Theorem 2] for the extended version. The asymptotic variances (4.2) and (4.3) can be estimated as follows:

$$ \begin{aligned} \hat{\sigma}_{\kappa}^2(\hat{F}) &= \left(\frac{\hat{\mu}_2}{\hat{\mu}_1^2} - 1\right) - \frac{2}{\kappa} \left(\frac{\hat{\mu}_{\kappa+1}}{\hat{\mu}_1\hat{\mu}_{\kappa}} - 1\right) + \frac{1}{\kappa^2} \left(\frac{\hat{\mu}_{2\kappa}}{\hat{\mu}_{\kappa}^2} - 1\right), \\ \hat{\sigma}^2(\hat{J}, \hat{F}) &= \sum_{i=1}^{n-1} \left(1 - \frac{i}{n}\right) \hat{J}\left(\frac{i}{n}\right) \left[ 2 \sum_{j=1}^{i-1} \frac{j}{n} \hat{J}\left(\frac{j}{n}\right) (X_{i+1,n} - X_{i,n})(X_{j+1,n} - X_{j,n}) \right. \\ & \qquad \left. + \frac{i}{n} \hat{J}\left(\frac{i}{n}\right) (X_{i+1,n} - X_{i,n})^2 \right], \end{aligned} \quad (4.4) $$

with for all $u \in (0, 1]$, $\hat{J}(u) = 1/\hat{\mu}_1 - g(1-u)/\widehat{D}_{1,g}(\mathcal{X}_n)$ and where $\widehat{D}_{1,g}(\mathcal{X}_n)$, $\hat{\mu}_{\kappa}$ are defined respectively in (2.2), (2.4).

Let us comment on the assumptions: (1) specifies the behaviour of $g'$ at 0, this is an usual condition when dealing with L-statistics, see for instance [38, Theorem 1]. Assumption (2) controls the tail heaviness of the distribution, ensuring in particular that the asymptotic variance of the estimator is finite. It is automatically fulfilled when $\gamma \le 0$, i.e. when $F$ is in the maximum domain of attraction of Gumbel or Weibull, see the proof of Example 2 in Appendix B.2.

It has already been stated that the weight functions associated with empirical inequality indices $\hat{I}_{\text{Gini}}(\mathcal{X}_n)$, $\hat{I}_{\text{Atkinson},\eta}(\mathcal{X}_n)$, $\hat{I}_{\text{EG},\nu}(\mathcal{X}_n)$, $\hat{I}_{\text{Bonferroni},1}(\mathcal{X}_n)$ and $\hat{I}_{\text{Mehran}}(\mathcal{X}_n)$ satisfy $(\mathbf{H}_2)$ and $(\mathbf{H}_4)$. In addition, for $g = g_{\nu,1}$ with $\nu > 1$, associated with Gini and extended Gini indices, it is clear that $g(0) = 0$, $\beta = 1$ and Assumption (1) holds with $s = 2$ and any $\delta \in (0, \min(1/4, \nu - 1))$. Similarly, for $g(u) = \log(1/(1-u))$ associated with the Bonferroni index ($h = 1$), it is easily seen that $g(0) = 0$, $\beta = 1$ and Assumption (1) holds with $s = 2$ and any $\delta \in (0, 1/4)$.

We conclude this paragraph by showing that the three distributions considered in Example 1 fulfil the assumptions of the above asymptotic normality theorem.
**Example 2** (Asymptotic normality of five inequality indices for three distributions). Let $\kappa \in (-\infty, 0) \cup (0, 1]$.

(i) If $X_\gamma \sim \text{Pareto}(1, \gamma)$ with $\gamma < 1/2$,

(ii) or $Y_\theta \sim \text{Beta}(\theta, 1)$ with $\theta > 0$ and $\kappa > -\theta/2$,

(iii) or $Z_b \sim \text{Weibull}(1, b)$ with $b > 0$ and $\kappa > -b/2$,

then, $\mu_2$ and $\mu_{2\kappa}$ are finite and Assumption (2) of Theorem 3 is satisfied for $(\beta, s) = (1, 2)$: $\hat{I}_{\text{Gini}}(\mathcal{X}_n)$, $\hat{I}_{\text{Atkinson},\eta}(\mathcal{X}_n)$, $\hat{I}_{\text{EG},\nu}(\mathcal{X}_n)$, $\hat{I}_{\text{Bonferroni},1}(\mathcal{X}_n)$ and $\hat{I}_{\text{Mehran}}(\mathcal{X}_n)$ are asymptotically Gaussian.

## 4.2 Estimation and asymptotic normality, grouped data case

Suppose that, for all $j \in \{1, \dots, m\}$, $F^{[j]}$ is the c.d.f. of the uniform distribution on $[a_j, a_{j+1})$ such that $0 = a_1 < \dots < a_{m+1} < \infty$. Then, the result of Proposition 3(i) can be simplified and allows us to define, under the same assumptions, an alternative estimator to (2.3) in the case of grouped frequency distributions:

$$ \widetilde{I}_{\kappa,g}(\mathcal{X}_n) = 1 - \frac{1}{\hat{\mu}_1} \left( \sum_{j=1}^{m} \hat{p}^{[j]} \int_0^1 [(a_{j+1} - a_j)u + a_j]^\kappa g \left( \hat{p}^{[j]}(1-u) + \sum_{i=j+1}^{m} \hat{p}^{[i]} \right) du \right)^{1/\kappa}, \quad (4.5) $$

where

$$ \hat{\mu}_1 = \sum_{j=1}^{m} \hat{p}^{[j]} \mu^{[j]} \quad \text{and for all } j \in \{1, \dots, m\}: \quad \hat{p}^{[j]} = \frac{1}{n} \sum_{i=1}^{n} \mathbb{1}_{[a_j, a_{j+1})}(X_i) \quad \text{and} \quad \mu^{[j]} = (a_j + a_{j+1})/2. $$

For example, in the particular case of the Gini index, simple calculations yield the estimator:

$$ \tilde{I}_{\text{Gini}}(\mathcal{X}_n) = 1 - \frac{1}{3\hat{\mu}_1} \sum_{j=1}^{m} \hat{p}^{[j]} \left( 3(a_{j+1} + a_j) \sum_{i=j+1}^{m} \hat{p}^{[i]} + (a_{j+1} + 2a_j)\hat{p}^{[j]} \right). \quad (4.6) $$

Note that another estimator of the Gini index dedicated to grouped data is provided in [24], based on the interpolation of the Lorenz curve using Hermite polynomials. Here, $\widetilde{I}_{\kappa,g}$ relies on a uniformity assumption within the groups (inherent in histogram representations) which may be interpreted as a quadratic interpolation of the Lorenz curve. It can reveal useful for estimating income inequality when individual data are unavailable. Its asymptotic normality is established as follows, letting $p = (p^{[1]}, \dots, p^{[m]})^\top$ and $\hat{p} = (\hat{p}^{[1]}, \dots, \hat{p}^{[m]})^\top$. First, the multivariate central limit theorem entails that

$$ \sqrt{n}(\hat{p} - p) \xrightarrow{n \to +\infty} \mathcal{N}(0, \Sigma(p)), $$

where $\Sigma(p) = \text{diag}(p) - pp^\top$ is a $m \times m$ matrix. Second, rewriting $\widetilde{I}_{\kappa,g}(\mathcal{X}_n) = \phi_{\kappa,g}(\hat{p})$, where for all $v \in (0, 1)^m$,

$$ \phi_{\kappa,g}(v) = 1 - \frac{1}{\sum_{j=1}^{m} v_j \mu^{[j]}} \left( \sum_{j=1}^{m} v_j \int_0^1 [(a_{j+1} - a_j)u + a_j]^\kappa g \left( v_j(1-u) + \sum_{i=j+1}^{m} v_i \right) du \right)^{1/\kappa}, $$

the delta method yields the asymptotic normality of $\widetilde{I}_{\kappa,g}(\mathcal{X}_n)$:

$$ \sqrt{n}(\widetilde{I}_{\kappa,g}(\mathcal{X}_n) - I_{\kappa,g}(X)) \xrightarrow{n \to +\infty} \mathcal{N}(0, \nabla\phi_{\kappa,g}(p)^\top \Sigma(p) \nabla\phi_{\kappa,g}(p)). $$

The asymptotic variance is too complicated to be reproduced there, but it can be easily estimated by replacing the unknown parameters by their empirical counterparts in the above equation. Both estimators (2.3) and (4.5) of the Gini index are compared in Subsection 6.2 on a real data set.
# 5 Illustration on simulated data

The goal of this section is to illustrate Theorem 1 which states that, for any value $I \in (0, 1)$ of an inequality index $I_{\kappa,g}$, there exists a distribution in each of the three maximum domains of attraction such that $I_{\kappa,g} = I$. To this end, let us consider the inequality indices presented in Subsection 2.2: $I_{\text{Gini}}$ (2.5), $I_{\text{Atkinson},\eta}$ (2.6) with $\eta = 0.9$, $I_{\text{EG},\nu}$ (2.7) with $\nu = 3$ which coincides with $I_{\text{Mehran}}$ (2.9) and $I_{\text{Bonferroni},h}$ (2.8) with $h = 1$. The target value is fixed to $I = 0.15$ and three distributions are used: Pareto(1, $\gamma$) $\in$ MDA(Fr\'echet), Beta($\theta$, 1) $\in$ MDA(Weibull) and Weibull(1, $b$) $\in$ MDA(Gumbel), see Table 1 in Appendix C and Example 2.

Using the expressions of the inequality indices associated with these three distributions provided in Table 2 in Appendix C, one can determine the parameters $\gamma$, $\theta$ and $b$ such that the inequality index is $I = 0.15$. For instance, in the case of the Gini index, one has $\gamma = 2I/(1 + I)$, $\theta = (1 - I)/(2I)$ and $b = -\log(2)/\log(1 - I)$. Closed-form expressions are not available for the three other indices, but Theorem 1 ensures that some solutions exist. These are approximated numerically using the function unireot from the R software [44].

For each inequality index, a sample of size $n = 1000$ is simulated $N = 1000$ times from each Pareto, Beta and Weibull distributions with the appropriate parameters depending on $I$. For each distribution, the vector of $N$ estimates of $\hat{I}_{\kappa,g}(\mathcal{X}_n)$ (see (2.3)) is computed and summarised as a violin plot on Figure 1. The empirical mean of the estimates is depicted by a solid vertical line. Note that the estimators of Gini and Atkinson indices are available in the R package *ineq* and are particular cases of (2.3), see [53]. The extreme-value index associated with each situation is superimposed to the graphs ($\gamma = 0$ for the Weibull distribution, $\gamma = -1$ for the Beta distribution and $\gamma \in \{0.21, 0.22, 0.26, 0.48\}$ for the Pareto distribution) highlighting the property established in Theorem 1: Distributions with very different tail behaviours may share the same inequality index.

As expected, all estimated mean values are close to the theoretical index $I = 0.15$ depicted by a dotted vertical line and shared by all three distributions. It however appears that the estimator of the Atkinson index (top right panel) suffers from a large variance when applied to a Pareto distribution. This is consistent with the asymptotic variance (4.2) in Theorem 3(i): Combining $\hat{I}_{\text{Atkinson},\eta}(\mathcal{X}_n)$ with $X_\gamma \sim \text{Pareto}(1, \gamma)$ of c.d.f. $F_\gamma$ yields

$$ \sigma_{1-\eta}^2(F_\gamma) = \frac{2\eta^2\gamma^4}{(1-2\gamma)(1-(2-\eta)\gamma)(1-2(1-\eta)\gamma)} \rightarrow +\infty \text{ as } \gamma \rightarrow (1/2)^-. $$

Here, $\gamma \approx 0.48$ and $\eta = 0.9$ yield an estimated asymptotic variance of $\hat{\sigma}_{1-\eta}^2(\hat{F}_\gamma) \approx 5.22$. As a comparison, the asymptotic variances of the Atkinson index associated with the Beta and Weibull distributions are approximately 0.1 and 0.06, respectively.

The densities of the three asymptotic Gaussian distributions (4.1) are superimposed on Figure 2. Their shape is consistent with the finite sample behaviour of $\hat{I}_{\text{Atkinson},0.9}(\mathcal{X}_n)$ depicted in the top right panel of Figure 1.

# 6 Illustrations on an income data set

A real data set is used in Subsection 6.1 to illustrate Theorem 2 and to compare estimators $\hat{I}_{\text{Gini}}$ and $\tilde{I}_{\text{Gini}}$ in Subsection 6.2: The U.S. National Medical Expenditure Survey<fnref n="3" /> (NMES) is a nationally representative study of health care use and costs. Let us consider the data set from the survey conducted from 1987 to 1988, and referred to as NMES1988 in the *AER* package [35] of the R software. It consists of $n = 4406$ households, all covered by Medicare, a public health insurance program. The focus here is on the income (in USD 10,000), which is depicted by a histogram in the top left panel of Figure 3.

<footnote n="3">³https://www.icpsr.umich.edu/web/NACDA/series/45</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1: Violin plots computed on $N = 1000$ replications of four estimated inequality indices $\hat{I}_{\text{Gini}}, \hat{I}_{\text{Atkinson},0.9}, \hat{I}_{\text{EG},3} = \hat{I}_{\text{Mehran}}$ and $\hat{I}_{\text{Bonferroni},1}$ on samples of size $n = 1000$ from three distributions with common theoretical inequality index $0.15 = I \in \{I_{\text{Gini}}, I_{\text{Atkinson},0.9}, I_{\text{EG},3} = I_{\text{Mehran}}, I_{\text{Bonferroni},1}\}$: in blue $X_{\gamma} \sim \text{Pareto}(1, \gamma)$, in green $Y_{\theta} \sim \text{Beta}(\theta, 1)$ and in red $Z_{b} \sim \text{Weibull}(1, b)$. Top left: Gini index, top right: Atkinson index with $\eta = 0.9$, bottom left: extended Gini index with $\nu = 3$ and bottom right: Bonferroni index with $h = 1$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2: Asymptotic Gaussian distributions from Theorem 3(i) applied to $\hat{I}_{\text{Atkinson},0.9}(\mathcal{X}_n)$ with three distributions: in blue $X_{\gamma} \sim \text{Pareto}(1, \gamma)$, in green $Y_{\theta} \sim \text{Beta}(\theta, 1)$ and in red $Z_{b} \sim \text{Weibull}(1, b)$.</figcaption></figure>
## 6.1 Illustration of the non-discrimination of upper tails

The aim here is to illustrate Theorem 2, which states that for any pair $(I_{\kappa,g}, \gamma)$ of inequality and extreme-value indices respectively, there exists an associated distribution. The pair $(\hat{I}_{\text{Gini}}, \hat{\gamma})$ estimated on the data set is our starting point for constructing two distributions with $I_{\text{Gini}} = \hat{I}_{\text{Gini}}$ and two extreme-value indices different from $\hat{\gamma}$.

The first step is to estimate the Gini index of the income distribution. The estimator (2.3) yields $\hat{I}_{\text{Gini}} \approx 0.470$.

The second step is to estimate the extreme-value index of the data distribution. To this aim, the Hill plot $k \in \{1, \dots, \lfloor n/4 \rfloor\} \mapsto H(k)$ is displayed in the top right panel of Figure 3, where $H$ denotes the Hill estimator [29]. The use of the Hill estimator is motivated by the histogram, which suggests that the underlying data distribution may be heavy-tailed. The parameter $k$ is selected using an algorithm based on the minimisation of the asymptotic mean squared error, which is implemented as the Hill.kopt function in the ReIns package [45], see [1, Section 4.2.1.1]. The pair $(k^{\text{opt}}, \hat{\gamma} = H(k^{\text{opt}})) = (366, 0.38)$ is represented by a red triangle on the Hill plot. As a visual check, a Pareto-quantile plot of $\log(X_{n-i+1,n}/X_{n-k^{\text{opt}},n})$ against $\log((k^{\text{opt}} + 1)/i)$ for $i \in \{1, \dots, k^{\text{opt}}\}$ is drawn in the bottom left panel of Figure 3. Comparing with the blue line of slope 0.38, an approximately linear relationship is observed and provides an empirical evidence that the data distribution belongs to MDA(Fréchet) with extreme-value index $\gamma \approx 0.38$.

The third step is to exhibit two distributions sharing the same Gini index $\hat{I}_{\text{Gini}} \approx 0.470$ with extreme-value indices different from $\hat{\gamma} = 0.38$, say $\gamma_1 = 0.3$ and $\gamma_2 = 0.5$. The construction of these two distributions is the same as in the proof of Theorem 2 (see Appendix B.2) in the case $\gamma \in (0, 1)$ and the same notations are thus adopted. Let $\gamma \in \{\gamma_1, \gamma_2\}$ and consider a mixture with proportion $p \in (0, 1)$ of a Beta($\theta$, 1) distribution ($\theta > 0$) and a Pareto(1, $\gamma$) distribution. Using Tables 1 and 2 in Appendix C and Proposition 3(iii) with $\kappa = 1$ yields

$$I_{\text{Gini}}(X_{\theta}^{[p]}) = 1 - \frac{2}{p\theta/(\theta+1) + (1-p)/(1-\gamma)} \left( \frac{p^2\theta^2}{(\theta+1)(2\theta+1)} + \frac{p(1-p)\theta}{\theta+1} + \frac{(1-p)^2}{2-\gamma} \right).$$

Solving the equation $I_{\text{Gini}}(X_{\theta}^{[p]}) = \hat{I}_{\text{Gini}}$ with respect to $p \in (0, 1)$ is equivalent to finding the roots of the second-degree polynomial

$$p \in (0, 1) \mapsto \mathcal{P}_{\theta, \gamma}(p) = a(\theta, \gamma)p^2 + b(\theta, \gamma)p + c(\gamma),$$

where

$$a(\theta, \gamma) = \frac{2(\gamma\theta + 1)}{(2\theta + 1)(2 - \gamma)},$$

$$b(\theta, \gamma) = \frac{\hat{I}_{\text{Gini}}(\gamma - 2)(\theta\gamma + 1) + \gamma^2\theta + 3\gamma - 2}{(\theta + 1)(2 - \gamma)(1 - \gamma)},$$

$$c(\gamma) = \frac{(2 - \gamma)\hat{I}_{\text{Gini}} - \gamma}{(2 - \gamma)(1 - \gamma)}.$$

Theorem 2 ensures that, for all $\theta > 0$ and $\gamma \in \{\gamma_1, \gamma_2\}$, the equation $\mathcal{P}_{\theta, \gamma}(p) = 0$ has at least one root $p_0(\theta, \gamma) \in (0, 1)$. This can be checked by considering $\Delta(\theta, \gamma)$ the associated discriminant with particular case $\Delta(0, \gamma) = (1 - \hat{I}_{\text{Gini}})^2/(1 - \gamma)^2$ yielding two roots:

$$p_0(0, \gamma) = \frac{(2 - \gamma)\hat{I}_{\text{Gini}} - \gamma}{2(1 - \gamma)} \text{ and } \tilde{p}_0(0, \gamma) = 1.$$

Moreover, $\gamma \in \{\gamma_1, \gamma_2\}$ implies $\hat{I}_{\text{Gini}} > \gamma/(2 - \gamma)$ leading to $p_0(0, \gamma) \in (0, 1)$. The continuity of $\theta \mapsto \Delta(\theta, \gamma)$ and $\theta \mapsto p_0(\theta, \gamma)$ on $[0, +\infty)$ entails the existence of $V_0$, a right neighbourhood of 0 such that, for all $\theta \in V_0$, $\mathcal{P}_{\theta, \gamma}$ has at least one root $p_0(\theta, \gamma) \in (0, 1)$. Numerically, one can pick $\theta_1 = 0.6$ leading to $p_0(\theta_1, \gamma_1) \approx 0.68$
and $\theta_2 = 1$ leading to $p_0(\theta_2, \gamma_2) \approx 0.52$. As expected, the resulting mixtures exhibit different tail densities, see the bottom right panel of Figure 3, despite they share the same Gini index.

<figure><img src="image_5.png" /><figcaption>Figure 3: Illustration on an income data set. Top left: histogram of the data set, top right: Hill plot with ($k^{\text{opt}}$, $\hat{\gamma} = H(k^{\text{opt}})$) = (366, 0.38) depicted by a red triangle, bottom left: Pareto-quantile plot with a line of slope $\hat{\gamma} = 0.38$, bottom right: Tail densities of two Beta-Pareto mixtures with extreme-value index $\gamma_1 = 0.3$ in blue and $\gamma_2 = 0.5$ in red. See text for details.</figcaption></figure>

## 6.2 Comparison of individual and grouped data estimators of the Gini index

Let us illustrate numerically the accuracy of the Gini estimator for grouped frequency distributions (4.6). Keeping in mind the notations of Subsection 4.2, the incomes are divided into $m = 48$ groups, following the same partitioning {$a_1, \dots, a_{m+1}$} as in the histogram displayed in the top left panel of Figure 3. This yields $\tilde{I}_{\text{Gini}} \approx 0.471$, recalling that $\hat{I}_{\text{Gini}} \approx 0.470$. To get an idea of the quality of this estimate, a 90% asymptotic confidence interval for $I_{\text{Gini}}$ is derived from Theorem 3(ii):

$$IC_{0.9,n}(I_{\text{Gini}}) = \left[ \frac{\hat{I}_{\text{Gini}} - u_{0.95}\hat{\sigma}(\hat{J}, \hat{F})/\sqrt{n}}{1 - u_{0.95}\hat{\sigma}(\hat{J}, \hat{F})/\sqrt{n}} ; \frac{\hat{I}_{\text{Gini}} + u_{0.95}\hat{\sigma}(\hat{J}, \hat{F})/\sqrt{n}}{1 + u_{0.95}\hat{\sigma}(\hat{J}, \hat{F})/\sqrt{n}} \right],$$

where $u_{0.95}$ denotes the 0.95 quantile of the standard Gaussian distribution and $\hat{\sigma}(\hat{J}, \hat{F})$ is defined in (4.4) with $g(u) = 2u$. Here, $IC_{0.9,n}(I_{\text{Gini}}) = [0.459; 0.480]$ includes $\tilde{I}_{\text{Gini}}$. Confidence intervals for the Gini index may also be computed using bootstrap methods [14, 40]. For instance, a 90% confidence interval obtained via bootstrap, as implemented in the Gini function (used with default parameters) of the DescTools package [48] from the R software, is [0.460; 0.481] on the considered data. This interval nearly coincides with the asymptotic version $IC_{0.9,n}(I_{\text{Gini}})$.

To conclude, these results indicate that the grouped data estimator provides an accurate approximation of the Gini index in this case.
# 7 Conclusion

This work provides a unified theoretical framework for a broad family of inequality indices, including most of the classical ones and satisfying six axioms. Two estimators of these indices are provided: One dedicated to individual data, and the other to grouped data. Their asymptotic normality is established and estimators of the associated asymptotic variances are proposed. Besides, we prove that all indices in this family share the non-discrimination property between and within maximum domains of attraction, thus limiting their sensitivity to extreme income values. These theoretical results are supported by simulations and illustrated on a real income data set.

This work is dedicated to normalised inequality indices (in the unit interval) and thus excludes for instance the Theil index [51] and the coefficient of variation [41]. A possible avenue for research would be to include these measures in a wider family and extend our theoretical results to this framework.

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Acknowledgements.

S. Girard acknowledges the support of the Chair “Stress Test, Risk Management and Financial Steering”,
led by the French Ecole Polytechnique and its Foundation and sponsored by BNP Paribas.

A Appendix: Axiomatic framework for inequality indices

The starting point to define an inequality index is a function *I* associating a non negative value with a
distribution, see [10, Appendix A.2]. An inequality index can be equivalently defined in terms of random
variables provided that *X* ≜ *Y* implies *I*(*X*) = *I*(*Y*). This convention is adopted hereafter. The following
six axioms are usually considered in the literature:

(A1) Normalisation. *I*(*X*) ∈ [0,1] where *I*(*X*) = 0 means perfect equality and *I*(*X*) = 1 perfect inequality in the underlying distribution of *X*, see [2, p.250].

(A2) Scale-invariance/income scale independence. Multiplying incomes of the whole population by the same positive factor does not change the initial index: ∀λ > 0, I(λX) = I(X), see [2, p.251] and [36, Property (6)].

(A3) **Aggregativity/decomposability principle.** The inequality index is computable without knowing the exact income distribution within subgroups, relying only on subgroup inequality levels and their aggregate characteristics. A formal definition based on the sample version of *I* is stated in [10, Appendix A.5.1] and [8, Definition 1]. The following population version is considered. Let $m \in \mathbb{N} \setminus \{0\}$. For all $j \in \{1, \dots, m\}$, let $X^{[j]}$ be a random variable with c.d.f. $F^{[j]}$, expectation $\mu^{[j]}$ and $p^{[j]} \in (0,1)$ such that $\sum_{j=1}^m p^{[j]} = 1$. Let $X$ be a random variable with associated c.d.f. $F = \sum_{j=1}^m p^{[j]} F^{[j]}$. An inequality index $I$ satisfies the aggregativity principle if there exists some function $S$ such that $I(X) = S(p^{[1]}, \dots, p^{[m]}, \mu^{[1]}, \dots, \mu^{[m]}, I(X^{[1]}), \dots, I(X^{[m]}))$.

(A4) Transfer principle/Pigou-Dalton condition. The inequality index decreases when a small amount of income is transferred from a richer to a poorer person, other things remaining the same, see [11, Section 4], [20, p.27] and [42, p.24]. This axiom can be stated using $\hat{I}$, the sample version of $I$. Let $\mathcal{X}_n = \{X_1, \dots, X_n\}$ be a sample of independent random variables with same distribution as $X$. Let $r \in \{2, \dots, n\}$, $h \in \{1, \dots, r-1\}$ and $\delta > 0$ such that $\delta < \min(X_{r-h+1,n} - X_{r-h,n}, X_{r,n} - X_{r-1,n})$. The modified sample (after transfer) $\mathcal{X}_n^{[\delta]}$ is defined as follows:

$$
X_{i,n}^{[\delta]} = \begin{cases} X_{r-h,n} + \delta & \text{if } i = r-h, \\ X_{r,n} - \delta & \text{if } i = r, \\ X_{i,n} & \text{otherwise.} \end{cases}
$$

Then,

$$
\lim_{\delta \to 0^+} \frac{\hat{I}(\mathcal{X}_n^{[\delta]}) - \hat{I}(\mathcal{X}_n)}{\delta} < 0 \text{ a.s.}
$$

(A5) Anonymity principle. The inequality index value does not depend on who earns which income, see [36, Property (3)]. This axiom can be formalised using $\hat{I}$, the sample version of $I$: Let $\mathcal{X}_n = \{X_1, \dots, X_n\}$ be a sample of independent random variables with same distribution as $X$. Let $\tau$ be a permutation of $\{1, \dots, n\}$ and denote by $\mathcal{X}_n^{(\tau)} = \{X_{\tau(1)}, \dots, X_{\tau(n)}\}$. Then, $\hat{I}(\mathcal{X}_n) = \hat{I}(\mathcal{X}_n^{(\tau)})$.

(A6) Population principle. The measured inequality in a population with *n* individuals remains the same when merging this population with another identical one, so that the new population has 2*n* individuals [47, Assumption (6)]: $\hat{I}(\mathcal{X}_n \cup \mathcal{X}_n) = \hat{I}(\mathcal{X}_n)$.
# B Appendix: Proofs

## B.1 Preliminary results

The first lemma provides an example of density $g$ satisfying $(\mathbf{H}_2)-(\mathbf{H}_4)$, which is repeatedly used in our work.

**Lemma 1.** The density of the Beta($\alpha, \beta$) distribution (2.1) satisfies $(\mathbf{H}_2)-(\mathbf{H}_4)$ for all $(\alpha, \beta) \in \mathcal{B}$ and

$$ \mathcal{K}(g_{\alpha,\beta}) = \begin{cases} (-\infty, 0) \cup (0, 1] & \text{if } \alpha > 1, \\ (-\infty, 0) \cup (0, 1) & \text{if } \alpha = 1. \end{cases} $$

*Proof of Lemma 1.* It is clear that $g_{\alpha,\beta}$ is a density function, defined on $[0, 1)$ and continuously differentiable on $(0, 1)$. One has for all $u \in (0, 1)$,

$$ g'_{\alpha,\beta}(u) = \frac{u^{\alpha-2}(1-u)^{\beta-2}}{B(\alpha, \beta)} [(2-(\alpha+\beta))u + \alpha - 1]. $$

The sign of $g'_{\alpha,\beta}$ is that of the function $l_{\alpha,\beta}(u) = (2-(\alpha+\beta))u + \alpha - 1$. Three cases appear.

- If $\alpha + \beta = 2$, then $l_{\alpha,\beta}(u) = \alpha - 1 \ge 0$ for all $u \in (0, 1)$ by assumption. The result follows.

- If $\alpha + \beta < 2$, then $l_{\alpha,\beta}$ is strictly increasing on $(0, 1)$ and $l_{\alpha,\beta}(0) = \alpha - 1 \ge 0$ by assumption. Thus, $l_{\alpha,\beta}$ is positive on $(0, 1)$.

- If $\alpha + \beta > 2$, then $l_{\alpha,\beta}$ is strictly decreasing on $(0, 1)$ and $l_{\alpha,\beta}(1) = 1 - \beta \ge 0$ by assumption. Thus, $l_{\alpha,\beta}$ is positive on $(0, 1)$ and $(\mathbf{H}_2)$ holds. Moreover, it is easily seen that $(\mathbf{H}_3)$ holds. Taking $x > 1 \mapsto L(x) = (1 - 1/x)^{\alpha-1}/B(\alpha, \beta)$, which is slowly-varying, yields $g_{\alpha,\beta}(u) = (1 - u)^{\beta-1}L(1/(1-u))$. Since $x^{-1}L(x) \sim B(\alpha, \beta)^{-1}x^{-1}$ as $x \to +\infty$, $(\mathbf{H}_4)$ also holds. Finally, $g_{\alpha,\beta}$ is strictly increasing and $g_{\alpha,\beta}(0) = 0$ as soon as $\alpha > 1$, which yields $\mathcal{K}(g_{\alpha,\beta})$. $\square$

The next lemma establishes the stability of $(\mathbf{H}_2)-(\mathbf{H}_4)$ with respect to some simple transformations.

**Lemma 2.** Suppose $(\mathbf{H}_2)-(\mathbf{H}_4)$ hold for $g^{[0]}$ and $g^{[1]}$. Then, for all $p \in [0, 1]$, $g^{[p]} = pg^{[1]} + (1-p)g^{[0]}$ satisfies $(\mathbf{H}_2)-(\mathbf{H}_4)$.

*Proof of Lemma 2.* First, as a convex combination $g^{[p]} = pg^{[1]} + (1-p)g^{[0]}$ is continuously differentiable, positive, increasing and integrates to one on $(0, 1)$. The function $g^{[p]}$ thus satisfies $(\mathbf{H}_2)$. Second, there exist $(\alpha_0, \alpha_1) \in [1, +\infty)^2$ and $(c_0, c_1) \in [0, +\infty)^2$ such that for all $u$ small enough, $g^{[0]}(u) \le c_0u^{\alpha_0-1}$ and $g^{[1]}(u) \le c_1u^{\alpha_1-1}$. It is then easy to see that $g^{[p]}(u) \le (c_0 + c_1)u^{\alpha_0\wedge\alpha_1-1}$ for all $u$ small enough. Thus, $(\mathbf{H}_3)$ holds for $g^{[p]}$ with $\alpha = \alpha_0 \wedge \alpha_1 \ge 1$. Third, there exist $(\beta_0, \beta_1) \in (0, 1]^2$ and $(L_0, L_1) \in \mathcal{RV}_0^2$ such that for all $u \in [0, 1]$, $g^{[j]}(u) = (1-u)^{\beta_j-1}L_j(1/(1-u))$, $j \in \{0, 1\}$. Suppose $\beta_0 < \beta_1$, without loss of generality. Then, $g^{[p]}(u) = (1-u)^{\beta_0-1}L_p(1/(1-u))$ with $L_p(t) = (1-p)L_0(t) + pt^{\beta_0-\beta_1}L_1(t)$, $t > 0$. Moreover, $L_1/L_0 \in \mathcal{RV}_0$ and $\beta_0 < \beta_1$ lead to $L_p(t) \sim (1-p)L_0(t)$ as $t \to +\infty$, from [6, Proposition 1.3.6]. It follows that $L_p \in \mathcal{RV}_0$ and $\int^{+\infty} L_p(t)/tdt$ diverges to $+\infty$. Thus, $g^{[p]}$ satisfies $(\mathbf{H}_4)$. $\square$

Lemma 3 below showcases a property of the c.d.f. associated with a density satisfying $(\mathbf{H}_2)$ that will reveal useful to prove Lemma 4.

**Lemma 3.** Suppose $(\mathbf{H}_2)$ holds. Then, the associated c.d.f. $G$ of $g$ satisfies $G(u) \le u$, for all $u \in [0, 1]$.

*Proof of Lemma 3.* Let $\phi$ be the function defined by $\phi(u) = G(u) - u$, for all $u \in (0, 1)$. It is differentiable on $(0, 1)$ and one has $\phi'(u) = g(u) - 1$ which is increasing because $g$ is. First, the aim is to show that there exists $u_1 \in (0, 1)$ such that $g(u_1) = 1$. Suppose, reasoning by contradiction, that for all $u \in (0, 1)$, $g(u) \ne 1$. If $g(0) = 1$, then $g(u) \ge 1$ for all $u \in (0, 1)$ as $g$ is increasing. But, since it is a continuous density on $(0, 1)$, it leads to $g \equiv 1$, which is a contradiction. Reasoning as above, if $g(u) \to 1$ as $u \to 1^-$, then $g \equiv 1$ and it is
a contradiction. Since g is continuous on $[0, 1)$, $g(u) < 1$ or $g(u) > 1$ for all $u \in [0, 1]$, by the intermediate value theorem. Then, $\int_0^1 g(u)du \neq 1$ and it is a contradiction. As a conclusion, there exists $u_1 \in (0, 1)$ such that $g(u_1) = 1$. Since $\phi'$ is increasing, one has $\phi'(0) \le \phi'(u_1) = 0 \le \phi'(1)$. As a consequence, $\phi$ is decreasing on $(0, u_1)$ and increasing on $(u_1, 1)$. Finally, since G is a c.d.f., $\phi(0) = \phi(1) = 0$ and it leads to $\phi(u) \le 0$ for all $u \in [0, 1]$. The result is thus proved. $\square$

The next lemma establishes the existence of $I_{\kappa,g}(X)$ under (H$_1$) and (H$_2$) when $\kappa \in \mathcal{K}(g)$.

**Lemma 4.** Suppose (H$_1$) and (H$_2$) hold. Then:

(i) For all $\kappa \in \mathcal{K}(g)$, $D_{\kappa,g}(X)$ exists in $[0, +\infty]$. Moreover, if $\kappa > 0$, then $D_{\kappa,g}(X) < \infty$.

(ii) For all $\kappa \in \mathcal{K}(g)$, $D_{\kappa,g}^{1/\kappa}(X) \le \mu$.

*Proof of Lemma 4.* Introduce the c.d.f. $\Psi = \bar{G} \circ \bar{F}$ and denote by $\mathbb{E}_{\Psi}$ the associated expectation. First, remark that $\mathbb{E}_{\Psi}(X)$ exists in $(0, +\infty]$ and

$$0 \le \mathbb{E}_{\Psi}(X) = \int_{0}^{+\infty} \bar{\Psi}(x)dx = \int_{0}^{+\infty} G(\bar{F}(x))dx \le \int_{0}^{+\infty} \bar{F}(x)dx = \mu, \quad (B.1)$$

in view of $X \ge 0$ a.s., (H$_2$) and Lemma 3.

(i) Since $X^\kappa \ge 0$ a.s., $\mathbb{E}_{\Psi}(X^\kappa)$ exists in $(0, +\infty]$ and one has

$$\mathbb{E}_{\Psi}(X^{\kappa}) = \int_{0}^{+\infty} x^{\kappa} \Psi'(x)dx = \int_{0}^{+\infty} x^{\kappa} g(\bar{F}(x))f(x)dx = \int_{0}^{1} q(u)^{\kappa} g(u)du = D_{\kappa,g}(X),$$

using the change of variable $u = \bar{F}(x)$. It remains to prove that, if $\kappa \in (0, 1]$, then $D_{\kappa,g}(X) < \infty$. Note that $q(u)^\kappa \to x_1^\kappa < \infty$ as $u \to 1^-$ and thus $u \mapsto q(u)^\kappa$ is bounded in a neighbourhood of $1^-$.

– Suppose first $x_\infty < \infty$. Then $q(u)^\kappa \to x_\infty^\kappa < \infty$ as $u \to 0^+$ and therefore $u \mapsto q(u)^\kappa$ is bounded in a neighbourhood of $0^+$. As a conclusion, since g is integrable on $(0, 1)$, it is also the case for $u \mapsto q(u)^\kappa g(u)$.

– Second, suppose $x_\infty = +\infty$. Then, there exists $\varepsilon > 0$ such that $0 \le u < \varepsilon$ implies $q(u) > 1$ which yields $q(u)^\kappa \le q(u)$, as $\kappa \le 1$. Since g is increasing on $[0, 1)$ and positive, one has $g(0) < \infty$ such that g is bounded in a neighbourhood of $0^+$. It yields $q(u)^\kappa g(u) \le Cq(u)$ in this neighbourhood of $0^+$, where $C > 0$. In view of (H$_1$), $\mu < \infty$ so $u \mapsto q(u)$ is integrable on $(0, 1)$. Thus, it is also the case for $u \mapsto q(u)^\kappa g(u)$. As a conclusion, $D_{\kappa,g}(X) < \infty$.

(ii) Two cases must be separated according to the sign of $\kappa$.

– Suppose first that $\kappa \in (0, 1]$. It has just been proved that $\mathbb{E}_{\Psi}(X^\kappa) < \infty$. Jensen inequality can be applied and the concavity of $0 < x \mapsto x^\kappa$ leads to $\mathbb{E}_{\Psi}(X^\kappa) \le \mathbb{E}_{\Psi}^{\kappa}(X) \le \mu^\kappa$, in view of (B.1). The result thus follows: $D_{\kappa,g}^{1/\kappa}(X) \le \mu$.

– Second, let us consider the case where $\kappa < 0$. If $\mathbb{E}_{\Psi}(X^\kappa) = +\infty$, then $\mathbb{E}_{\Psi}^{1/\kappa}(X^\kappa) = 0 \le \mu$. Otherwise, $\mathbb{E}_{\Psi}(X^\kappa) < \infty$ and the convexity of $0 < x \mapsto x^\kappa$ implies $\mathbb{E}_{\Psi}(X^\kappa) \ge \mathbb{E}_{\Psi}^{\kappa}(X)$ so that $D_{\kappa,g}^{1/\kappa}(X) = \mathbb{E}_{\Psi}^{1/\kappa}(X^\kappa) \le \mathbb{E}_{\Psi}(X) \le \mu$ in view of (B.1). $\square$

The following lemma ensures that (H$_2$)-(H$_4$) hold for the weight function associated with the class of Bonferroni indices, see Subsection 2.2.

**Lemma 5.** For all $h \ge 1$, $g(u) = h \int_{1-u}^1 (1-p)^{h-1}/p \, dp$ satisfies (H$_2$)-(H$_4$).

*Proof of Lemma 5.* First, it is clear that g is continuously differentiable and positive on $(0, 1)$. Moreover, Fubini's theorem yields

$$\int_{0}^{1} g(u)du = h \int_{0}^{1} (1-p)^{h-1}dp = 1.$$
For all $u \in (0, 1)$, $g'(u) = hu^{h-1}/(1-u) > 0$ which ensures that $g$ is strictly increasing. Thus, (H$_2$) holds. Second, since $g(0) = 0$, (H$_3$) holds with $\alpha = 1$. Third, one has $g(u) = L(1/(1-u))$ with for all $t \ge 1$,

$$L(t) = h \int_{1/t}^{1} \frac{(1-p)^{h-1}}{p} dp.$$

Besides, since $h \int_{0}^{1} (1-p)^{h-1}/p \, dp$ diverges to $+\infty$ and $h(1-p)^{h-1}/p \sim h/p$ as $p \to 0^+$, $L(t) \sim h \log(t)$ as $t \to +\infty$, in view of [28, Theorem 3 p.159] and thus $L \in \mathcal{RV}_0$. Finally, for all $t \ge 1$, $L(t)/t \ge (1-1/t)^h/t$, which yields $\int_{+\infty}^{+\infty} L(t)/t \, dt = +\infty$ and (H$_4$) thus holds with $\beta = 1$. $\square$

The following lemma provides the extreme-value index associated with some mixture, it is used to prove Lemma 7. We refer to [9, Section 4] for a similar result: When both components of the mixture satisfy (C$_1$), the extreme-value index of the mixture is the largest of the two indices (max-functional property).

**Lemma 6.** Let $F^{[1]}$ and $F^{[0]}$ be two c.d.f. with endpoint $x_{\infty}^{[1]}$ and $x_{\infty}^{[0]}$ respectively. If $x_{\infty}^{[1]} < x_{\infty}^{[0]}$ and $F^{[0]}$ satisfies (C$_1$) with extreme-value index $\gamma \in \mathbb{R}$ and auxiliary function $a(\cdot)$, then the c.d.f. $F^{[p]} = pF^{[1]} + (1-p)F^{[0]}$ satisfies (C$_1$) for all $p \in [0, 1)$ with same extreme-value index and auxiliary function.

*Proof of Lemma 6.* The c.d.f. $F^{[0]}$ satisfies (C$_1$) with extreme-value index $\gamma$, so equivalently, by [12, Theorem 1.1.6], for all $x$ such that $1 + \gamma x > 0$:

$$\lim_{t \to x_{\infty,0}^{[0]}} \frac{\bar{F}^{[0]}(t + xh^{[0]}(t))}{\bar{F}^{[0]}(t)} = (1 + \gamma x)^{-1/\gamma}, \quad (B.2)$$

where $\bar{F}^{[0]} = 1 - F^{[0]}$ and $h^{[0]}(\cdot) = a(1/\bar{F}^{[0]}(\cdot))$. The goal is now to prove that $\bar{F}^{[p]} = 1 - F^{[p]}$ also satisfies (B.2), with same $h^{[0]}$ and $\gamma$. Since $x_{\infty}^{[1]} < x_{\infty}^{[0]}$, the endpoint of $F^{[p]}$ is $x_{\infty}^{[0]}$. Letting $t \in \mathbb{R}$ and $x$ such that $1 + \gamma x > 0$, one has:

$$\frac{\bar{F}^{[p]}(t + xh^{[0]}(t))}{\bar{F}^{[p]}(t)} = \frac{p\bar{F}^{[1]}(t + xh^{[0]}(t)) + (1-p)\bar{F}^{[0]}(t + xh^{[0]}(t))}{p\bar{F}^{[1]}(t) + (1-p)\bar{F}^{[0]}(t)}.$$

Remarking that (B.2) implies $\bar{F}^{[0]}(t+xh^{[0]}(t)) \to 0$ as $t \to x_{\infty}^{[0]}$, it follows that $t+xh^{[0]}(t) \to x_{\infty}^{[0]}$ as $t \to x_{\infty}^{[0]}$. Moreover, $\bar{F}^{[1]}(t) = 0$ for all $t \ge x_{\infty}^{[1]}$. Thus, there exists $T \in [x_{\infty}^{[1]}, x_{\infty}^{[0]})$ such that for all $t \in [T, x_{\infty}^{[0]}):$

$$\bar{F}^{[1]}(t + xh^{[0]}(t)) = \bar{F}^{[1]}(t) = 0.$$

Then, one has

$$\lim_{t \to x_{\infty}^{[0]}} \frac{\bar{F}^{[p]}(t + xh^{[0]}(t))}{\bar{F}^{[p]}(t)} = \lim_{t \to x_{\infty}^{[0]}} \frac{\bar{F}^{[0]}(t + xh^{[0]}(t))}{\bar{F}^{[0]}(t)} = (1 + \gamma x)^{-1/\gamma}.$$

So equivalently, in view of [12, Theorem 1.1.6], $F^{[p]}$ satisfies (C$_1$) with auxiliary function $a$ and extreme-value index $\gamma$. The result is thus proved. $\square$

Lemma 7 below establishes that, under some assumptions, any extreme-value index can be associated with any inequality index $I_{\kappa,g}$. It will reveal useful to prove Theorem 2.

**Lemma 7.** Let $\gamma \in \mathbb{R}$ and $I \in (0, 1)$. Let $F^{[0]}$ be a c.d.f. and consider $(\bar{F}_{\theta}^{[1]})_{\theta \in \Theta}$, $\Theta \subset \mathbb{R}$, a set of c.d.f. where $\theta$ is a parameter of the distribution. Let $X_{\theta}^{[1]} \sim F_{\theta}^{[1]}$ and $X^{[0]} \sim F^{[0]}$. Suppose they satisfy (H$_1$) and denote respectively their endpoints by $x_{\infty}^{[1,\theta]}$ and $x_{\infty}^{[0]}$. Let $g$ be a density function satisfying (H$_2$) and let $\kappa \in \mathcal{K}(g)$. Suppose the three assumptions below are satisfied:

(1) For all $\theta \in \Theta$, $x_{\infty}^{[1,\theta]} < x_{\infty}^{[0]}$ and $F^{[0]}$ satisfies (C$_1$) with extreme-value index $\gamma$.

(2) The mapping $\theta \mapsto I_{\kappa,g}(X_{\theta}^{[1]})$ is a surjection from $\Theta$ onto $(0, 1)$.
(3) For all $\theta \in \Theta$, $\phi_{\theta}: p \mapsto I_{\kappa,g}(X_{\theta}^{[p]})$ with $X_{\theta}^{[p]} \sim pF_{\theta}^{[1]} + (1-p)F^{[0]}$ is continuous on $(0,1)$.

Then, there exist $\theta_{\star} \in \Theta$ and $p_{\star} \in (0,1)$ such that $\phi_{\theta_{\star}}(p_{\star}) = I$. Moreover, $X_{\theta_{\star}}^{[p_{\star}]}$ satisfies (C1) with extreme-value index $\gamma \le 1$ and $I_{\kappa,g}(X_{\theta_{\star}}^{[p_{\star}]}) = I$.

Note that Assumption (3) is referred to as the mixture-continuous property in [9, Definition 3.6].

**Proof of Lemma 7.** First, assumption (H1) on $X^{[0]}$ imposes $\mathbb{E}(X^{[0]}) < \infty$ which necessarily sets the restriction $\gamma \le 1$, under Assumption (1) of Lemma 7. Since $X^{[0]}$ and $X_{\theta}^{[1]}$ satisfy (H1) for all $\theta \in \Theta$, $g$ satisfies (H2) and $\kappa \in \mathcal{K}(g)$, then $\phi_{\theta}$ is well defined on $[0,1]$ for all $\theta \in \Theta$, see Proposition 3(i). Two cases appear.

- Suppose $I < I_{\kappa,g}(X^{[0]}$). Assumption (2) of Lemma 7 provides the existence of $\theta_{\star} \in \Theta$ such that $I_{\kappa,g}(X_{\theta_{\star}}^{[1]}) < I$. Then, $\phi_{\theta_{\star}}(1) = I_{\kappa,g}(X_{\theta_{\star}}^{[1]}) < I < I_{\kappa,g}(X^{[0]}) = \phi_{\theta_{\star}}(0)$.

- Suppose $I > I_{\kappa,g}(X^{[0]}$). Assumption (2) of Lemma 7 provides the existence of $\theta_{\star} \in \Theta$ such that $I_{\kappa,g}(X_{\theta_{\star}}^{[1]}) > I$. Then, $\phi_{\theta_{\star}}(0) = I_{\kappa,g}(X^{[0]}) < I < I_{\kappa,g}(X_{\theta_{\star}}^{[1]}) = \phi_{\theta_{\star}}(1)$.

In view of Assumption (3) of Lemma 7, the intermediate value theorem shows that, in both cases, there exists $p_{\star} \in (0,1)$ such that $\phi_{\theta_{\star}}(p_{\star}) = I_{\kappa,g}(X_{\theta_{\star}}^{[p_{\star}]}) = I$. In view of Assumption (1) of Lemma 7, Lemma 6 can be applied and thus the extreme-value index of $X_{\theta_{\star}}^{[p_{\star}]} \sim p_{\star}F_{\theta_{\star}}^{[1]} + (1-p_{\star})F^{[0]}$ is $\gamma$. $\square$

The next two lemmas provide sufficient conditions to fulfil the assumptions of [38, Theorem 1] dedicated to the asymptotic normality of L-statistics. They will be needed to prove Theorem 3.

**Lemma 8.** Suppose (H2) and (H4) hold, $g(0) = 0$ and $\beta \in (1/2, 1]$. Assume moreover there exist $s \ge 2$ and $\delta \in (0, \beta/2 - 1/4)$ such that $g'(u) = \mathcal{O}(u^{-3/2+1/s+\delta})$ as $u \to 0^+$. Then, there exists $M > 0$ such that for all $u \in (0,1)$,

$$|g(1-u)| \le Mu^{1/r-1/2}(1-u)^{1/s-1/2}, \qquad (B.3)$$

$$|g'(1-u)| \le Mu^{-3/2+1/r+\delta}(1-u)^{-3/2+1/s+\delta}, \qquad (B.4)$$

where $r = 4/(2\beta - 1) \ge 2$.

**Proof of Lemma 8.** Let us introduce for the sake of simplicity $\tilde{g}(\cdot) = g(1 - \cdot)$ and remark that $|\tilde{g}'(\cdot)| = g'(1 - \cdot)$. Our first goal is to prove (B.3). It is clear that $u \mapsto \tilde{g}(u)$ and $u \mapsto u^{1/r-1/2}(1-u)^{1/s-1/2}$ are both continuous on $(0,1)$. Since $\tilde{g}(1) = 0$, there exists a neighbourhood $V_1 \subset (0,1)$ of $1^{-}$ where $\tilde{g}$ is bounded. Moreover, $\tilde{g}(u) = u^{\beta-1}L(1/u)$ in view of (H4) and $u^{-1/r+\beta-1/2}L(1/u) \to 0$ as $u \to 0^+$ since $\beta > 1/2$, from [6, Proposition 1.3.6]. Consequently, there exists a neighbourhood $V_0 \subset (0,1)$ of $0^+$ where $u \mapsto u^{1/2-1/r}\tilde{g}(u)$ is bounded. Combining the above two results proves that there exists $M_1 > 0$ such that (B.3) holds with $M = M_1$.

Second, let us similarly prove (B.4). It is clear that $\tilde{g}'$ and $u \mapsto u^{-3/2+1/r+\delta}(1-u)^{-3/2+1/s+\delta}$ are continuous on $(0,1)$. Since $g'(u) = \mathcal{O}(u^{-3/2+1/s+\delta})$ as $u \to 0^+$, there exists a neighbourhood $\tilde{V}_1 \subset (0,1)$ of $1^{-}$ where $u \mapsto (1-u)^{3/2-1/s-\delta}|\tilde{g}'(u)|$ is bounded. Besides, (3.2) yields $|\tilde{g}'(u)| = u^{\beta-2}\tilde{L}(1/u)$ with $\tilde{L} \in \mathcal{RV}_0$. Moreover, since $\delta < \beta/2 - 1/4$, [6, Proposition 1.3.6] ensures that $u^{3/2-1/r-\delta}|\tilde{g}'(u)| \to 0$ as $u \to 0^+$. It thus exists a neighbourhood $\tilde{V}_0 \subset (0,1)$ of $0^+$ where $u \mapsto u^{3/2-1/r-\delta}|\tilde{g}'(u)|$ is bounded. Combining the above two points entails the existence of $M_2 > 0$ such that (B.3) holds with $M = M_2$.

Finally, taking $M = \max(M_1, M_2)$ proves the result. $\square$

**Lemma 9.** Let $s \ge 2$. Suppose (H1) and (C1) hold with $\gamma < 1/s$. Then,

$$\int^{+\infty} \bar{F}^{1/s}(x)dx < \infty. \qquad (B.5)$$
Proof of Lemma 9. Suppose first $\gamma > 0$. Since $F \in \text{MDA}(\text{Fréchet})$, then $\bar{F} \in \mathcal{RV}_{-1/\gamma}$ in view of [12, Theorem 1.2.1]. Moreover, $\gamma < 1/s$ and [6, Proposition 1.5.10] imply (B.5). Second, suppose $\gamma < 0$. It is clear that (B.5) holds since $x_{\infty} < \infty$. Finally, suppose $\gamma = 0$. Two cases appear. If $x_{\infty} < \infty$, then (B.5) holds for the same reason as for $\gamma < 0$. Otherwise, [16, Corollary 3.3.32] ensures that $\bar{F}$ and therefore also $\bar{F}^{1/s}$ are rapid-varying, see [16, Definition A3.11]. Then, [16, Theorem A3.12] implies (B.5), since $\bar{F}^{1/s}$ is decreasing. The result is thus proved. $\square$

## B.2 Proofs of main results

*Proof of Proposition 1.* Since $X \ge 0$ a.s. and $g$ is positive by (H1) and (H2), it is clear that $I_{\kappa,g}(X) \le 1$. The positivity of $I_{\kappa,g}(X)$ is a direct consequence of Lemma 4(ii) and the normalisation thus holds.

The scale-invariance follows from Definition 1 since, for any $\lambda > 0$, the quantile function and expectation of the random variable $\lambda X$ are respectively $\lambda q$ and $\lambda \mu$, leading to $I_{\kappa,g}(\lambda X) = I_{\kappa,g}(X)$. $\square$

*Proof of Proposition 2.* First, let us address the transfer principle and introduce

$$ \Delta \widehat{D}_{\kappa,g}(\delta) = (\widehat{D}_{\kappa,g}(\mathcal{X}_n^{[\delta]}) - \widehat{D}_{\kappa,g}(\mathcal{X}_n))/\delta, $$

$$ \Delta \widehat{I}_{\kappa,g}(\delta) = (\widehat{I}_{\kappa,g}(\mathcal{X}_n^{[\delta]}) - \widehat{I}_{\kappa,g}(\mathcal{X}_n))/\delta. $$

The aim is to prove that $\lim_{\delta \to 0^+} \Delta \widehat{I}_{\kappa,g}(\delta) < 0$. To this end, remark that $\hat{\mu}(\mathcal{X}_n^{[\delta]}) = \hat{\mu}(\mathcal{X}_n)$, and it is thus sufficient to focus on $\Delta \widehat{D}_{\kappa,g}(\delta)$. Simple calculations yield

$$ \Delta \widehat{D}_{\kappa,g}(\delta) = \frac{1}{\delta} \left\{ \left[ (X_{r,n} - \delta)^{\kappa} - X_{r,n}^{\kappa} \right] \int_{\frac{n-r}{n}}^{\frac{n-r+1}{n}} g(u) du + \left[ (X_{r-h,n} + \delta)^{\kappa} - X_{r-h,n}^{\kappa} \right] \int_{\frac{n-r+h}{n}}^{\frac{n-r+h+1}{n}} g(u) du \right\}. $$

Two first-order Taylor expansions at $\delta = 0$ entail

$$ \lim_{\delta \to 0^+} \Delta \widehat{D}_{\kappa,g}(\delta) = \kappa \left[ X_{r-h,n}^{\kappa-1} \int_{\frac{n-r+h}{n}}^{\frac{n-r+h+1}{n}} g(u) du - X_{r,n}^{\kappa-1} \int_{\frac{n-r}{n}}^{\frac{n-r+1}{n}} g(u) du \right]. $$

The change of variable $v = u - h/n$ in the first above integral leads to

$$ \lim_{\delta \to 0^+} \Delta \widehat{D}_{\kappa,g}(\delta) = \kappa \int_{\frac{n-r}{n}}^{\frac{n-r+1}{n}} \left[ X_{r-h,n}^{\kappa-1} g(u+h/n) - X_{r,n}^{\kappa-1} g(u) \right] du. $$

Two cases appear:

- If $\kappa = 1$, (H2) ensures that $g$ is strictly increasing which yields for all $u \in ((n-r)/n, (n-r+1)/n)$, $g(u+h/n) > g(u)$, as $h > 0$. Then, $\lim_{\delta \to 0^+} \Delta \widehat{D}_{1,g}(\delta) > 0$ and therefore $\lim_{\delta \to 0^+} \Delta \widehat{I}_{1,g}(\delta) < 0$ a.s.

- If $\kappa < 1$, then the simple increasingness of $g$ given by (H2) and $\kappa - 1 < 0$ ensure that $\lim_{\delta \to 0^+} \Delta \widehat{D}_{\kappa,g}(\delta)$ is almost surely of the same sign as $\kappa$. Moreover, the function $t \mapsto 1 - t^{1/\kappa}/\mu$ is strictly decreasing if $\kappa > 0$ and strictly increasing if $\kappa < 0$. In both cases, $\lim_{\delta \to 0^+} \Delta \widehat{I}_{\kappa,g}(\delta) < 0$ a.s. and the transfer principle is thus satisfied.

Second, let us focus on the anonymity principle. The sample mean is invariant with respect to permutations: $\hat{\mu}_1 = \frac{1}{n} \sum_{i=1}^n X_{\tau(i)}$. Moreover, the ordered sample is also invariant with respect to permutations of the original one: $\{X_{1,n}, \dots, X_{n,n}\} = \{X_{\tau(1),n}, \dots, X_{\tau(n),n}\}$ so that $\widehat{D}_{\kappa,g}(\mathcal{X}_n) = \widehat{D}_{\kappa,g}(\mathcal{X}_n^{(\tau)})$. As a conclusion, $\widehat{I}_{\kappa,g}(\mathcal{X}_n)$ satisfies the anonymity principle.

Finally, let us consider the population principle. It is clear that the sample mean of $\mathcal{X}_n \cup \mathcal{X}_n = \{X_{1,n}, X_{1,n}, \dots, X_{n,n}, X_{n,n}\} =: \{Y_{1,2n}, \dots, Y_{2n,2n}\}$ is also $\hat{\mu}_1$. Remarking that for all odd $j \in \{1, \dots, 2n\}$,
$Y_{2n-j+1,2n} = Y_{2n-j,2n} = X_{n-(j-1)/2,n}$, it follows

$$
\begin{align*}
\widehat{D}_{\kappa,g}(\mathcal{X}_n \cup \mathcal{X}_n) &= \sum_{j=1}^{2n} Y_{2n-j+1,2n}^{\kappa} [G(j/(2n)) - G((j-1)/(2n))] \\
&= \sum_{\text{odd } j=1}^{2n} Y_{2n-j+1,2n}^{\kappa} [G((j+1)/(2n)) - G((j-1)/(2n))],
\end{align*}
$$

and the change of variable $i = (j - 1)/2 + 1$ yields

$$
\widehat{D}_{\kappa,g}(\mathcal{X}_n \cup \mathcal{X}_n) = \sum_{i=1}^{n} X_{n-i+1,n}^{\kappa} [G(i/n) - G((i-1)/n)] = \widehat{D}_{\kappa,g}(\mathcal{X}_n).
$$

Consequently, $\widehat{I}_{\kappa,g}(\mathcal{X}_n \cup \mathcal{X}_n) = \widehat{I}_{\kappa,g}(\mathcal{X}_n)$ and the population principle holds.

<figure><img src="image_6.png" /></figure>

Proof of Proposition 3. For all $j \in \{1, \dots, m\}$, the density function associated with $X^{[j]}$ is denoted by $f^{[j]}$, its starting point is denoted by $x_1^{[j]}$ and its endpoint by $x_\infty^{[j]}$. Since these $m$ supports are non overlapping, suppose $x_1^{[1]} < \dots < x_1^{[m]}$, without loss of generality.

(i) It is clear that $X$ is continuous and a.s. positive. Since $\mu^{[j]}$ is finite for all $j \in \{1, \dots, m\}$, it is also the case for $\mu$. Besides, $\overline{F}^{[j]}$ is strictly decreasing for all $j \in \{1, \dots, m\}$, so that $\bar{F} = \sum_{j=1}^{m} p^{[j]} \overline{F}^{[j]}$ is also strictly decreasing. Consequently, (H$_1$) holds for $X$. Moreover, the change of variable $x = q(u)$ yields

$$
\begin{align*}
D_{\kappa,g}(X) &= \int_0^1 q(u)^\kappa g(u) du \\
&= \int_0^{+\infty} x^\kappa g \left( \sum_{i=1}^m p^{[i]} \overline{F^{[i]}}(x) \right) \left( \sum_{j=1}^m p^{[j]} f^{[j]}(x) \right) dx \\
&= \sum_{j=1}^m p^{[j]} \int_{x_1^{[j]}}^{x_\infty^{[j]}} x^\kappa g \left( \sum_{i=1}^m p^{[i]} \overline{F^{[i]}}(x) \right) f^{[j]}(x) dx \\
&= \sum_{j=1}^m \int_{x_1^{[j]}}^{x_\infty^{[j]}} x^\kappa p^{[j]} g \left( p^{[j]} \overline{F^{[j]}}(x) + \sum_{i=j+1}^m p^{[i]} \right) f^{[j]}(x) dx \\
&= \sum_{j=1}^m \left[ G \left( \sum_{i=j}^m p^{[i]} \right) - G \left( \sum_{i=j+1}^m p^{[i]} \right) \right] \int_{x_1^{[j]}}^{x_\infty^{[j]}} x^\kappa g^{[j]}(\overline{F^{[j]}}(x)) f^{[j]}(x) dx \\
&= \sum_{j=1}^m \omega^{[j]} D_{\kappa,g^{[j]}}(X^{[j]}).
\end{align*}
$$

The result is a direct consequence of Definition 1. Besides, it is easily seen that $g^{[j]}$ satisfies (H$_2$) for all
$j \in \{1, \dots, m\}$.

(ii) Suppose there exists $g$ satisfying (H$_2$) such that $g^{[j]} = g$ for all $j \in \{1, \dots, m\}$ and $(p^{[1]}, \dots, p^{[m]}) \in (0, 1)^m$. In particular, $g(u) = g^{[1]}(u)$, for all $u \in [0, 1)$ and all $(p^{[1]}, \dots, p^{[m]}) \in (0, 1)^m$. Then, for $u = 0$ and letting $y = \sum_{i=j+1}^{m} p^{[i]}$ one has

$$
(1 - y) \frac{g(y)}{\bar{G}(y)} = g(0), \quad \forall y \in (0, 1).
$$

The solution of this differential equation is $u \mapsto g(0)(1-u)^{g(0)-1} = g_{1,g(0)}(u)$ with $0 < g(0) \le 1$ such that $g_{1,g(0)}$ is positive and increasing. Moreover, in particular, $g(u) = g^{[m]}(u)$, for all $u \in [0,1)$ and all $(p^{[1]}, \dots, p^{[m]}) \in (0,1)^m$. Following exactly the same steps, the only solution is $g = g_{g(1),1}$ with $g(1) \ge 1$. Since, necessarily $g^{[1]} = g^{[m]} = g$, the previous two points yield $g(0)(1-u)^{g(0)-1} = g(1)u^{g(1)-1}$, for all
$u \in [0, 1]$. Then, considering $u = 0$ leads to $g(0) = g(1) = 1$ or $g(0) = 0$ and $g(1) > 1$. In the first case, one has $g \equiv 1$, while in the second case, considering $u = 1$ entails $g(1) = 0$ which is not possible. As a consequence, the only solution satisfying (H$_2$) is $g \equiv 1$. Conversely, $g \equiv 1$ entails for all $j \in \{1, \dots, m\}$, all $(p^{[1]}, \dots, p^{[m]}) \in (0, 1)^m$ and all $u \in [0, 1]$,

$$g^{[j]}(u) = \frac{p^{[j]}}{\omega^{[j]}} g \left( p^{[j]}u + \sum_{i=j+1}^{m} p^{[i]} \right) = \frac{p^{[j]}}{\sum_{i=j}^{m} p^{[i]} - \sum_{i=j+1}^{m} p^{[i]}} = 1 = g(u).$$

Then, considering $g \equiv 1$ in Proposition 3(i) proves the result.

(iii) Let $g(u) = au$, with $u \in [0, 1]$ and $a > 0$. By (H$_2$), $g$ integrates to one on $(0, 1)$, which imposes $a = 2$ i.e. $g = g_{2,1}$. Moreover, for all $j \in \{1, \dots, m\}$, all $(p^{[1]}, \dots, p^{[m]}) \in (0, 1)^m$ and all $u \in [0, 1]$,

$$\omega^{[j]} = p^{[j]} \left( 2 \sum_{i=j+1}^{m} p^{[i]} + p^{[j]} \right) \quad \text{and} \quad g^{[j]}(u) = \frac{(p^{[j]})^2}{\omega^{[j]}} g(u) + \frac{2p^{[j]} \sum_{i=j+1}^{m} p^{[i]}}{\omega^{[j]}}.$$

A simple calculation leads to $I_{\kappa,g_{2,1}}(X)$ and the result is thus proved. $\square$

**Proof of Example 1.** (i) Definition 1 combined with assumptions on $(\alpha, \beta)$ and $\kappa$ lead to the result.

(ii) Suppose first $\kappa/\theta + \beta > 0$. Then, by assumptions on $(\alpha, \beta)$ and Definition 1, the result follows. Second, suppose $\kappa/\theta + \beta \le 0$. The aim is to prove that $D_{\kappa,g_{\alpha,\beta}}(Y_\theta) = \int_0^1 q(u)^\kappa g_{\alpha,\beta}(u)du$ diverges to $+\infty$. Let $s \in (0, 1)$. Then for all $u \in (s, 1)$,

$$q(u)^\kappa g_{\alpha,\beta}(u) = u^{\alpha-1}(1-u)^{\kappa/\theta+\beta-1} \ge s^{\alpha-1}(1-u)^{\kappa/\theta+\beta-1}.$$

Moreover, $\int_s^1 (1-u)^{\kappa/\theta+\beta-1} du$ diverges to $+\infty$ since $\kappa/\theta + \beta \le 0$. As a consequence, $D_{\kappa,g_{\alpha,\beta}}(Y_\theta)$ diverges to $+\infty$. Finally, remarking that $\kappa \le -\theta\beta < 0$, $D_{\kappa,g_{\alpha,\beta}}(Y_\theta)^{1/\kappa} = 0$ and the result is proved.

(iii) Lemma 4(i) ensures that $D_{\kappa,g_{\alpha,\beta}}(Z_b)$ exists in $[0, +\infty]$ and is finite if $\kappa > 0$. Definition 1 leads to

$$D_{\kappa,g_{\alpha,\beta}}(Z_b) = \frac{1}{B(\alpha, \beta)} \int_0^1 (\log(1/u))^{\kappa/b} u^{\alpha-1} (1-u)^{\beta-1} du$$

The aim is to prove that $D_{\kappa,g_{\alpha,\beta}}(Z_b)$ is finite if $\kappa/b + \beta > 0$ and diverges to $+\infty$ otherwise. Let $\kappa < 0$. It is clear that $u \mapsto (\log(1/u))^{\kappa/b} u^{\alpha-1} (1-u)^{\beta-1}$ is bounded in a neighbourhood of $0^+$. The change of variable $x = \log(1/u)$ yields

$$\int_{1/e}^{1} (\log(1/u))^{\kappa/b} u^{\alpha-1} (1-u)^{\beta-1} du = \int_{0}^{1} e^{-\alpha x} (1-e^{-x})^{\beta-1} x^{\kappa/b} dx.$$

Moreover, $e^{-\alpha x}(1-e^{-x})^{\beta-1}x^{\kappa/b} \sim x^{\kappa/b+\beta-1}$ as $x \to 0^+$. As a consequence, $D_{\kappa,g_{\alpha,\beta}}(Z_b)$ is of the same nature as $\int_0^1 x^{\kappa/b+\beta-1} dx$, which is finite if $\kappa/b + \beta > 0$ and diverges to $+\infty$ otherwise. Finally, in view of $\kappa < 0$, $D_{\kappa,g_{\alpha,\beta}}(Z_b)^{1/\kappa} = 0$, the result is thus obtained. $\square$

**Proof of Theorem 1.** (i) Consider the random variables defined for all $\gamma \in (0, 1)$ by $X_\gamma \sim \text{Pareto}(1, \gamma)$. These random variables clearly satisfy (H$_1$). By Definition 1 and Table 1 in Appendix C, one has

$$I_{\kappa,g}(X_\gamma) = 1 - (1 - \gamma) \left( \int_0^1 u^{-\kappa\gamma} g(u) du \right)^{1/\kappa}.$$

The aim is to prove that $\gamma \mapsto I_{\kappa,g}(X_\gamma)$ is a surjection from $(0, 1)$ onto $(0, 1)$. First, the mapping $\gamma \mapsto u^{-\kappa\gamma}g(u)$ is clearly continuous on $(0, 1)$, for all $u \in (0, 1)$. Moreover, for all $(u, \gamma) \in (0, 1)^2$, $u^{-\kappa\gamma}g(u) \le u^{-\kappa\gamma\vee0}g(u)$ for any $\kappa \in (-\infty, 0) \cup (0, 1]$. By (H$_2$), the function $g$ is integrable on $(0, 1)$. It is also the case of $u \mapsto u^{-\kappa}g(u)$ since (H$_3$) yields $u^{-\kappa}g(u) = \mathcal{O}(u^{\alpha-\kappa-1})$, with $\alpha > \kappa$ in view of the assumption
($\kappa, \alpha) \neq (1, 1)$. Thus, $\gamma \mapsto I_{\kappa,g}(X_\gamma)$ is continuous on $(0, 1)$. Second, $u^{-\kappa\gamma} \to 1$ as $\gamma \to 0^+$ and $u^{-\kappa\gamma} \to u^{-\kappa}$ as $\gamma \to 1^-$ for all $u \in (0, 1)$. Then, the dominated convergence theorem yields $I_{\kappa,g}(X_\gamma) \to 0$ as $\gamma \to 0^+$ and $I_{\kappa,g}(X_\gamma) \to 1$ as $\gamma \to 1^-$. Thus, by the intermediate value theorem, $\gamma \mapsto I_{\kappa,g}(X_\gamma)$ is a surjection from $(0, 1)$ onto $(0, 1)$. Finally, since $X_\gamma \in \text{MDA}(\text{Fréchet})$ for all $\gamma \in (0, 1)$, the result follows.

(ii) Consider the random variables defined by $Y_\theta \sim \text{Beta}(\theta, 1)$ for all $\theta \in \Theta_{\kappa,\beta} = \{\theta > 0 | \kappa/\theta + \beta > 0\}$, with $\beta \in (0, 1]$ from (H$_4$). This set of random variables clearly satisfies (H$_1$). By Definition 1 and Table 1 in Appendix C, one has

$$I_{\kappa,g}(Y_{\theta}) = 1 - \frac{\theta+1}{\theta} \left( \int_{0}^{1} (1-u)^{\kappa/\theta} g(u) du \right)^{1/\kappa} = 1 - \left( \int_{0}^{1} (1+1/\theta)^{\kappa} (1-u)^{\kappa/\theta} g(u) du \right)^{1/\kappa}.$$

The aim is to prove that $\theta \mapsto I_{\kappa,g}(Y_\theta)$ is a surjection from $\Theta_{\kappa,\beta}$ onto $(0, 1)$. Two cases appear, according to the sign of $\kappa$.

- Suppose $\kappa < 0$ leading to $\Theta_{\kappa,\beta} = (-\kappa/\beta, +\infty)$. First, $\theta \mapsto (1-u)^{\kappa/\theta}g(u)$ is clearly continuous on $\Theta_{\kappa,\beta}$, for all $u \in (0, 1)$. Let $\theta \in [\theta_1, \theta_2] \subset \Theta_{\kappa,\beta}$. Then, for all $u \in (0, 1)$, $(1-u)^{\kappa/\theta}g(u) \le (1-u)^{\kappa/\theta_1}g(u)$. Combining (H$_4$) and [6, Proposition 1.5.10] yields

$$\int_0^1 (1-u)^x g(u) du \quad \begin{cases} < +\infty & \text{if } x > -\beta, \\ = +\infty & \text{if } x \le -\beta. \end{cases} \qquad (B.6)$$

Then, since $\kappa/\theta_1 > -\beta$, $u \mapsto (1-u)^{\kappa/\theta_1}g(u)$ is integrable on $(0, 1)$ and $\theta \mapsto I_{\kappa,g}(Y_\theta)$ is thus continuous on $\Theta_{\kappa,\beta}$, by [18, Theorem 2.27]. Second, Fatou's lemma and (B.6) give $\int_0^1 (1-u)^{\kappa/\theta}g(u)du \to +\infty$ as $\theta \to (-\kappa/\beta)^+$. Since $\kappa < 0$, it proves that $I_{\kappa,g}(Y_\theta) \to 1$ as $\theta \to (-\kappa/\beta)^+$. Third, suppose $\theta \ge 1 - \kappa/\beta$. Then, for all $u \in (0, 1)$, $(1-u)^{\kappa/\theta}g(u) \le (1-u)^{\kappa\beta/(\beta-\kappa)}g(u)$, which is integrable on $(0, 1)$ by (B.6), as $\kappa\beta/(\beta - \kappa) > -\beta$. Moreover, $(1-u)^{\kappa/\theta}g(u) \to g(u)$ as $\theta \to +\infty$. The dominated convergence theorem thus yields $I_{\kappa,g}(Y_\theta) \to 0$ as $\theta \to +\infty$. By the intermediate value theorem, $\theta \mapsto I_{\kappa,g}(Y_\theta)$ is a surjection from $\Theta_{\kappa,\beta}$ onto $(0, 1)$.

- Suppose $\kappa > 0$ so that $\Theta_{\kappa,\beta} = (0, +\infty)$. First, $\theta \mapsto (1-u)^{\kappa/\theta}g(u)$ is clearly continuous on $\Theta_{\kappa,\beta}$, for all $u \in (0, 1)$. Moreover, for all $(u, \theta) \in (0, 1) \times \Theta_{\kappa,\beta}$, $(1-u)^{\kappa/\theta}g(u) \le g(u)$, which is integrable on $(0, 1)$ by (H$_2$). Then, $\theta \mapsto I_{\kappa,g}(Y_\theta)$ is continuous on $\Theta_{\kappa,\beta}$. Second, let us investigate the limit as $\theta \to 0^+$. Note that (H$_4$) combined with [6, Proposition 1.3.6] yields $(1-u)^xg(u) \to 0$ as $u \to 1^-$, for all $x > 1 - \beta$. Let $x > 1$ and suppose $\theta < \kappa/(x-1)$. Since $g$ is continuously differentiable by (H$_2$), an integration by parts is allowed and yields

$$\int_0^1 (1 + 1/\theta)^\kappa (1 - u)^{\kappa/\theta} g(u) du = \frac{(1 + 1/\theta)^\kappa}{1 + \kappa/\theta} \left( g(0) + \int_0^1 (1 - u)^x g'(u) (1 - u)^{1+\kappa/\theta-x} du \right). \quad (B.7)$$

For all $u \in (0, 1)$, $(1-u)^xg'(u)(1-u)^{1+\kappa/\theta-x} \to 0$ as $\theta \to 0^+$. Moreover, for all $u \in (0, 1)$,

$$(1-u)^x g'(u) (1-u)^{1+\kappa/\theta-x} \le (1-u)^x g'(u) \le g'(u).$$

By integration by parts, $u \mapsto (1-u)^xg'(u)$ is integrable in a neighbourhood of $1^-$. Since $g(0) < \infty$, $u \mapsto g'(u)$ is integrable in a neighbourhood of $0^+$. As a consequence, the dominated convergence theorem can be applied and ensures that the integral of the right-hand side of (B.7) tends to 0 as $\theta \to 0^+$. If $\kappa < 1$, the integral of the left-hand side of (B.7) tends to 0 as $\theta \to 0^+$. Now if $\kappa = 1$ i.e. $1 \in \mathcal{K}(g)$, it yields $g(0) = 0$ such that the same integral tends also to 0 as $\theta \to 0^+$. Thus, $I_{\kappa,g}(Y_\theta) \to 1$ as $\theta \to 0^+$. Third, the dominated convergence theorem leads easily to $I_{\kappa,g}(Y_\theta) \to 0$ as $\theta \to +\infty$. By the intermediate value theorem, $\theta \mapsto I_{\kappa,g}(Y_\theta)$ is a surjection from $\Theta_{\kappa,\beta}$ onto $(0, 1)$. Finally, since $Y_\theta \in \text{MDA}(\text{Weibull})$ for all $\theta \in \Theta_{\kappa,\beta}$, the result follows.

(iii) Two cases arise according to the sign of $\kappa$.
– Suppose first $\kappa < 0$. Consider the random variables $Z_b$ defined for all $b > 0$ by $Z_b - 1 \sim \text{Weibull}(1, b)$ which satisfy (H$_1$). Definition 1 leads to

$$I_{\kappa,g}(Z_b) = 1 - \frac{1}{1 + \Gamma(1 + 1/b)} \left( \int_{0}^{1} (1 + \log(1/u)^{1/b})^{\kappa} g(u) du \right)^{1/\kappa}.$$

The aim is to prove that $b \mapsto I_{\kappa,g}(Z_b)$ is a surjection from $(0, +\infty)$ onto $(0, 1)$. It is clear that $b \mapsto (1 + \log(1/u)^{1/b})^{\kappa} g(u)$ is continuous on $(0, +\infty)$ for all $u \in (0, 1)$. Moreover, since $1 + \log(1/u)^{1/b} \ge 1$ for all $u \in (0, 1)$, one has

$$(1 + \log(1/u)^{1/b})^{\kappa} g(u) \le g(u), \quad \forall u \in (0, 1). \qquad (B.8)$$

Consequently, since $u \mapsto g(u)$ is integrable on $(0, 1)$, $b \mapsto D_{\kappa,g}(Z_b)$ is continuous on $(0, +\infty)$ and so is $b \mapsto I_{\kappa,g}(Z_b)$. Let us investigate the limit as $b \to 0^+$ where two cases appear. If $u \in (1/e, 1)$, then $(1+\log(1/u)^{1/b})^{\kappa} \to 1$ as $b \to 0^+$. Otherwise, if $u \in (0, 1/e)$, then $(1+\log(1/u)^{1/b})^{\kappa} \to 0$ as $b \to 0^+$. In both cases, (B.8) allows to apply the dominated convergence theorem, which yields $D_{\kappa,g}(Z_b) \to 1 - G(1/e) \ne 0$ as $b \to 0^+$. Moreover, $\Gamma(1+1/b) \to +\infty$ as $b \to 0^+$ implies $I_{\kappa,g}(Z_b) \to 1$ as $b \to 0^+$. Now, for all $u \in (0, 1)$, $(1+\log(1/u)^{1/b})^{\kappa} \to 2^{\kappa}$ as $b \to +\infty$. With (B.8), the dominated convergence theorem can be applied and leads to $D_{\kappa,g}(Z_b) \to 2^{\kappa}$ as $b \to +\infty$ and thus to $I_{\kappa,g}(Z_b) \to 0$ as $b \to +\infty$. The mapping $b \mapsto I_{\kappa,g}(Z_b)$ is thus a surjection from $(0, +\infty)$ onto $(0, 1)$. Since $Z_b \in \text{MDA}(\text{Gumbel})$ for all $b > 0$, the result follows.

– Suppose now $\kappa > 0$ and consider the random variables defined for all $b > 0$ by $Z_b \sim \text{Weibull}(1, b)$, which satisfy (H$_1$). Definition 1 leads to

$$I_{\kappa,g}(Z_b) = 1 - \frac{1}{\Gamma(1 + 1/b)} \left( \int_{0}^{1} \log(1/u)^{\kappa/b} g(u) du \right)^{1/\kappa}.$$

The aim is to prove that $b \mapsto I_{\kappa,g}(Z_b)$ is a surjection from $(0, +\infty)$ onto $(0, 1)$. It is clear that $b \mapsto (\log(1/u))^{\kappa/b} g(u)$ is continuous on $(0, +\infty)$. Note that

$$(\log(1/u))^{\kappa/b} g(u) \le g(u), \quad \forall u \in (1/e, 1). \qquad (B.9)$$

Let us consider $b \in [b_1, b_2] \subset (0, +\infty)$. Then,

$$(\log(1/u))^{\kappa/b} g(u) \le (\log(1/u))^{\kappa/b_1} g(u), \quad \forall u \in (0, 1/e),$$

with $u \mapsto (\log(1/u))^{\kappa/b_1} g(u)$ integrable on $(0, 1)$ since it is the integrand of $D_{\kappa,g}(Z_{b_1})$, see Lemma 4. Thus, [18, Theorem 2.27] can be applied and ensures that $b \mapsto D_{\kappa,g}(Z_b)$ is continuous on $(0, +\infty)$ and so is $b \mapsto I_{\kappa,g}(Z_b)$. By (H$_3$), there exist $u_0 \in (0, 1/e)$ and $c > 0$ such that for all $u \in (0, u_0)$, $g(u) \le cu^{\alpha-1}$. Then,

$$\int_{0}^{u_0} (\log(1/u))^{\kappa/b} g(u) du \le \frac{c}{\alpha} \int_{\alpha \log(1/u_0)}^{+\infty} (x/\alpha)^{\kappa/b} e^{-x} dx \le \frac{c \Gamma(\kappa/b + 1)}{\alpha^{\kappa/b+1}},$$

by the change of variable $x = \alpha \log(1/u)$. It follows that, on the one hand:

$$\frac{1}{\Gamma(1/b + 1)^\kappa} \int_0^{u_0} (\log(1/u))^{\kappa/b} g(u) du \le \frac{\kappa c}{\alpha} \alpha^{-\kappa/b} b^{\kappa-1} \frac{\Gamma(\kappa/b)}{\Gamma(1/b)^\kappa} =: \phi_1(b).$$

Moreover, by Stirling's formula and the assumption $(\kappa, \alpha) \ne (1, 1)$, one has $\phi_1(b) \to 0$ as $b \to 0^+$. On the other hand,

$$\frac{1}{\Gamma(1/b + 1)^\kappa} \int_{u_0}^{1} \log(1/u)^{\kappa/b} g(u) du \le \left( \frac{b \log(1/u_0)^{1/b}}{\Gamma(1/b)} \right)^\kappa \int_{u_0}^{1} g(u) du =: \phi_2(b).$$
Again Stirling's formula yields $\phi_2(b) \to 0$ as $b \to 0^+$. Thus, as $\kappa > 0$, one has shown that $I_{\kappa,g}(Z_b) \to 1$ as $b \to 0^+$. Besides, it is clear that $(\log(1/u))^{\kappa/b}g(u) \to g(u)$ as $b \to +\infty$. Let $b \ge 1$. For all $u \in (0, 1/e)$, one has $(\log(1/u))^{\kappa/b}g(u) \le (\log(1/u))^{\kappa}g(u)$ which is integrable on $(0, 1)$ since it is the integrand of $D_{\kappa,g}(Z_1)$ with $\kappa > 0$, see Lemma 4. Combining this result with (B.9) and recalling that $g$ is integrable on $(0, 1)$, the dominated convergence theorem can be applied and leads to $I_{\kappa,g}(Z_b) \to 0$ as $b \to +\infty$. As a conclusion, $b \mapsto I_{\kappa,g}(Z_b)$ is a surjection from $(0, +\infty)$ onto $(0, 1)$. Finally, since $Z_b \in \text{MDA}(\text{Gumbel})$ for all $b > 0$, the result follows. $\square$

**Proof of Theorem 2.** Let $I \in (0, 1)$. The proof is divided into three parts.

– Suppose first $\gamma \in (0, 1)$. The goal is to apply Lemma 7. Let $X^{[0]} \sim F^{[0]}$ where $F^{[0]}$ is the c.d.f. of the Pareto$(1, \gamma)$ distribution, which satisfies (H1) and (C1) with extreme-value index $\gamma$. Let us consider the subset $\Theta_{\kappa,\beta}$ defined in the proof of Theorem 1(ii). For all $\theta \in \Theta_{\kappa,\beta}$, let $X_{\theta}^{[1]} \sim F_{\theta}^{[1]}$ where $F_{\theta}^{[1]}$ is the c.d.f. of the Beta$(\theta, 1)$ distribution, which also satisfies (H1). Lemma 6 ensures that Assumption (1) of Lemma 7 holds. Besides, the mapping $\theta \mapsto I_{\kappa,g}(X_{\theta}^{[1]})$ is a surjection from $\Theta_{\kappa,\beta}$ onto $(0, 1)$, in view of the proof of Theorem 1(ii). Assumption (2) of Lemma 7 is thus satisfied. For all $p \in (0, 1)$ and all $\theta \in \Theta_{\kappa,\beta}$, let $X_{\theta}^{[p]} \sim pF_{\theta}^{[1]} + (1-p)F^{[0]}$. Applying Proposition 3(i) and the two changes of variable $x = q_{\theta}^{[1]}(u)$ and $x = q^{[0]}(u)$, gives:

$$ \phi_{\theta}(p) = I_{\kappa,g}(X_{\theta}^{[p]}) = 1 - \frac{1}{\mu_{\theta}^{[p]}} (p\tilde{\phi}_{1,\theta}(p) + (1-p)\tilde{\phi}_{0}(p))^{1/\kappa}, \quad (B.10) $$

with

$$ \tilde{\phi}_{1,\theta}(p) = \int_{0}^{1} x^{\kappa} g \left( p \bar{F}_{\theta}^{[1]}(x) + 1 - p \right) f_{\theta}^{[1]}(x) dx, \quad \tilde{\phi}_{0}(p) = \int_{1}^{+\infty} x^{\kappa} g \left( (1-p) \bar{F}^{[0]}(x) \right) f^{[0]}(x) dx, $$

and $\mu_{\theta}^{[p]} = p\mathbb{E}(X_{\theta}^{[1]}) + (1-p)\mathbb{E}(X^{[0]}) > 0$. It is clear that $p \mapsto x^{\kappa}g(p\bar{F}_{\theta}^{[1]}(x) + 1-p)f_{\theta}^{[1]}(x)$ is continuous on $(0, 1)$ for all $\theta \in \Theta_{\kappa,\beta}$ and all $x \in (0, 1)$. For all $x > 1$, $p \mapsto x^{\kappa}g((1-p)\bar{F}^{[0]}(x))f^{[0]}(x)$ is also continuous on $(0, 1)$. For all $x > 1$ and all $p \in (0, 1)$, one has

$$ x^{\kappa} g \left( (1-p) \bar{F}^{[0]}(x) \right) f^{[0]}(x) \le x^{\kappa} g \left( \bar{F}^{[0]}(x) \right) f^{[0]}(x), $$

from (H2). This right-hand side is integrable on $(1, +\infty)$ since $D_{\kappa,g}(X^{[0]})$ is finite, in view of the proof of Theorem 1(i). Similarly, for all $p \in [p_1, p_2] \subset (0, 1)$ and all $x \in (0, 1)$,

$$ x^{\kappa} g \left( p \bar{F}_{\theta}^{[1]}(x) + 1 - p \right) f_{\theta}^{[1]}(x) = x^{\kappa} g \left( 1 - p F_{\theta}^{[1]}(x) \right) f_{\theta}^{[1]}(x) \le x^{\kappa} g \left( 1 - p_1 F_{\theta}^{[1]}(x) \right) f_{\theta}^{[1]}(x). $$

Moreover, by the change of variable $u = 1 - p_1 F_{\theta}^{[1]}(x)$, one has

$$ \int_{0}^{1} x^{\kappa} g \left( 1 - p_1 F_{\theta}^{[1]}(x) \right) f_{\theta}^{[1]}(x) dx = \frac{1}{p_1^{\kappa/\theta+1}} \int_{1-p_1}^{1} (1-u)^{\kappa/\theta} g(u) du, \quad (B.11) $$

which is finite since $\kappa/\theta > -\beta$, see (B.6) in the proof of Theorem 1(ii). Applying [18, Theorem 2.27], it follows that $\tilde{\phi}_{1,\theta}$ and $\tilde{\phi}_{0}$ are continuous on $(0, 1)$. As a conclusion, for all $\theta \in \Theta_{\kappa,\beta}$, $p \mapsto \phi_{\theta}(p)$ is continuous on $(0, 1)$. Consequently, Assumption (3) of Lemma 7 holds and applying Lemma 7 leads to the existence of $\theta_{\star} \in \Theta_{\kappa,\beta}$ and $p_{\star} \in (0, 1)$ such that the random variable $X_{\theta_{\star}}^{[p_{\star}]}$ with extreme-value index $\gamma$ satisfies $I_{\kappa,g}(X_{\theta_{\star}}^{[p_{\star}]}) = I$.

– Second, suppose $\gamma < 0$. The proof follows the same steps as above. Let us consider the same random variables $X_{\theta}^{[1]} \sim \text{Beta}(\theta, 1)$ with $\theta \in \Theta_{\kappa,\beta}$. Let $X^{[0]} \sim F^{[0]}$, the c.d.f. of the Beta$(1, -1/\gamma)$ distribution shifted by 1, so that its support is $(1, 2)$. It satisfies (H1) and (C1) with extreme-value index $\gamma$. Moreover by assumption, (H2) holds. Lemma 6 ensures that Assumption (1) of Lemma 7 holds.
As before, Assumption (2) of Lemma 7 is satisfied, in view of the proof of Theorem 1(ii). Using (B.10), the last point of Assumption (3) of Lemma 7 to be checked is the continuity of

$$p \mapsto \tilde{\phi}_0(p) = \int_1^2 x^\kappa g \left( (1-p)\bar{F}^{[0]}(x) \right) f^{[0]}(x)dx.$$

For all $x \in (1, 2)$, $p \mapsto x^\kappa g((1-p)\bar{F}^{[0]}(x)) f^{[0]}(x)$ is continuous on $(0, 1)$. Moreover, for all $x \in (1, 2)$ and all $p \in (0, 1)$, one has

$$x^{\kappa} g \left( (1-p) \bar{F}^{[0]}(x) \right) f^{[0]}(x) \leq x^{\kappa} g \left( \bar{F}^{[0]}(x) \right) f^{[0]}(x),$$

from (H2). The change of variable $u = \bar{F}^{[0]}(x)$ then yields

$$\int_1^2 x^\kappa g \left( \bar{F}^{[0]}(x) \right) f^{[0]}(x)dx = D_{\kappa,g}(X^{[0]}) = \int_0^1 (2 - u^{-\gamma})^\kappa g(u)du.$$

This integral is finite since, for all $u \in (0, 1)$, $(2 - u^{-\gamma})^\kappa < 2^{\kappa\gamma 0}$. By theorem, $p \mapsto \tilde{\phi}_0(p)$ is continuous on $(0, 1)$ and so is $p \mapsto \phi_\theta(p) = I_{\kappa,g}(X_\theta^{[p]})$, for all $\theta \in \Theta_{\kappa,\beta}$. As a conclusion, Assumption (3) of Lemma 7 holds and applying Lemma 7 leads to the existence of $\theta_* \in \Theta_{\kappa,\beta}$ and $p_* \in (0, 1)$ such that the random variable $X_{\theta_*}^{[p_*]}$ with extreme-value index $\gamma$ satisfies $I_{\kappa,g}(X_{\theta_*}^{[p_*]}) = I$.

– Finally, suppose $\gamma = 0$. Theorem 1(iii) yields the existence of a random variable $X \in \text{MDA}(\text{Gumbel})$ i.e. with extreme-value index $\gamma = 0$ such that $I_{\kappa,g}(X) = I$. □

**Proof of Theorem 3.** (i) We propose below an elementary proof, based on the central limit theorem. However, one can also remark that $\hat{I}_{\kappa,1}(\mathcal{X}_n)$ is a function of two U-statistics. As a consequence, [30, Theorem 7.5] allows to prove the same result, under different assumptions. Let us remark that $D_{\kappa,1}(X) = \mu_\kappa$ and $\tilde{D}_{\kappa,1}(\mathcal{X}_n) = \hat{\mu}_\kappa$. Note also that, in the case $\kappa > 0$, $\mu_2 < \infty$ implies that $\mu_\kappa, \mu_{\kappa+1}$ and $\mu_{2\kappa}$ are finite by Jensen inequality. Besides, in the case $\kappa < 0$, Jensen inequality ensures that $\mu_\kappa$ and $\mu_{\kappa+1}$ are finite since $\mu_2 < \infty$ and $\mu_{2\kappa} < \infty$ by assumption. The central limit theorem entails

$$Z_{\kappa,n} = \sqrt{n}(\hat{\mu}_{\kappa} - \mu_{\kappa}) \xrightarrow{n \to +\infty} \mathcal{N}(0, \text{Var}(X^{\kappa})) \text{ and } Z_{1,n} = \sqrt{n}(\hat{\mu}_1 - \mu) \xrightarrow{n \to +\infty} \mathcal{N}(0, \text{Var}(X)). \quad (B.12)$$

Two first-order Taylor expansions then yield

$$\frac{\hat{\mu}_{\kappa}^{1/\kappa}}{\hat{\mu}_1} = \frac{\mu_{\kappa}^{1/\kappa}}{\mu} \left( 1 + \frac{Z_{\kappa,n}}{\sqrt{n}\kappa\mu_{\kappa}} + \mathcal{O}\left(\frac{1}{n}\right) \right) \left( 1 - \frac{Z_{1,n}}{\sqrt{n}\mu} + \mathcal{O}\left(\frac{1}{n}\right) \right),$$

and it easily follows that

$$\sqrt{n}(\hat{I}_{\kappa,1}(\mathcal{X}_n) - I_{\kappa,1}(X)) = \sqrt{n} \left( \frac{\mu_{\kappa}^{1/\kappa}}{\mu} - \frac{\hat{\mu}_{\kappa}^{1/\kappa}}{\hat{\mu}_1} \right) = \frac{\mu_{\kappa}^{1/\kappa}}{\mu} \left( \frac{Z_{1,n}}{\mu} - \frac{Z_{\kappa,n}}{\kappa\mu_{\kappa}} \right) + \mathcal{O}\left(\frac{1}{\sqrt{n}}\right). \quad (B.13)$$

Moreover, taking account of

$$\frac{Z_{1,n}}{\mu} - \frac{Z_{\kappa,n}}{\kappa\mu_{\kappa}} = \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^{n} \left( \frac{X_i}{\mu} - \frac{X_i^{\kappa}}{\kappa\mu_{\kappa}} \right) - \left( 1 - \frac{1}{\kappa} \right) \right)$$

and recalling that $\mu_\kappa^{1/\kappa}/\mu = 1 - I_{\kappa,1}(X)$, the central limit theorem yields

$$\sqrt{n} \left( \frac{\hat{I}_{\kappa,1}(\mathcal{X}_n) - I_{\kappa,1}(X)}{1 - I_{\kappa,1}(X)} \right) \xrightarrow{n \to +\infty} \mathcal{N}(0, \sigma_{\kappa}^2(F)).$$

The result is thus proved.

(ii) Let us first focus on $\tilde{D}_{1,g}(\mathcal{X}_n)$ and introduce $\tilde{g}(\cdot) = g(1 - \cdot)$. One has

$$\tilde{D}_{1,g}(\mathcal{X}_n) = \frac{1}{n} \sum_{i=1}^{n} \tilde{g}\left(\frac{i}{n+1}\right) X_{i,n} + \sum_{i=1}^{n} X_{i,n} \int_{\frac{i-1}{n}}^{\frac{i}{n}} \left(\tilde{g}(u) - \tilde{g}\left(\frac{i}{n+1}\right)\right) du =: \tilde{D}_{1,g}^{[1]}(\mathcal{X}_n) + \tilde{D}_{1,g}^{[2]}(\mathcal{X}_n),$$
leading to

$$
Z_{g,n} = \sqrt{n}(\widehat{D}_{1,g}(\mathcal{X}_n) - D_{1,g}(X)) = \sqrt{n}(\widehat{D}_{1,g}^{[1]}(\mathcal{X}_n) - D_{1,g}(X)) + \sqrt{n}\widehat{D}_{1,g}^{[2]}(\mathcal{X}_n).
$$

The first step is to prove the asymptotic normality of $\sqrt{n}(\widehat{D}_{1,g}^{[1]}(\mathcal{X}_n) - D_{1,g}(X))$. Assumption (1) of Theorem 3 ensures that Lemma 8 can be applied to prove that conditions (B.3) and (B.4) hold for $\tilde{g}$. Moreover, in view of Assumption (2) of Theorem 3, Lemma 9 yields (B.5) which is equivalent to $\int_0^1 u^{1/r}(1-u)^{1/s}dF^{-1}(u) < \infty$, by the change of variable $x = F^{-1}(u)$. Consequently, standard results on L-statistics [38, Theorem 1] and [46, Example 1] show that

$$
\begin{align*}
\sqrt{n}(\widehat{D}_{1,g}^{[1]}(\mathcal{X}_n) - \mathbb{E}(\widehat{D}_{1,g}^{[1]}(\mathcal{X}_n))) &\xrightarrow[n \to +\infty]{} \mathcal{N}(0, \sigma^2(\tilde{g}, F)), \\
\sqrt{n}(\mathbb{E}(\widehat{D}_{1,g}^{[1]}(\mathcal{X}_n)) - D_{1,g}(X)) &\xrightarrow[n \to +\infty]{} 0,
\end{align*}
$$

where $\sigma^2(\tilde{g}, F) = \int_0^{+\infty} \int_0^{+\infty} \tilde{g}(F(x))\tilde{g}(F(y)) [F(\min(x, y) - F(x)F(y))]dxdy$. It readily follows that

$$
\sqrt{n}(\widehat{D}_{1,g}^{[1]}(\mathcal{X}_n) - D_{1,g}(X)) \xrightarrow[n \to +\infty]{} \mathcal{N}(0, \sigma^2(\tilde{g}, F)). \quad (B.14)
$$

Now, let us show that $\sqrt{n}\widehat{D}_{1,g}^{[2]}(\mathcal{X}_n) \xrightarrow[n \to +\infty]{} 0$. Remark that the first term ($i=1$) of $\widehat{D}_{1,g}^{[2]}(\mathcal{X}_n)$ can be written as

$$
\Delta_n = \frac{X_{1,n}}{n+1} \tilde{g} \left( \frac{1}{n+1} \right) \int_0^2 \left( \frac{\tilde{g}(s/(n+1))}{\tilde{g}(1/(n+1))} \right) - 1 \right) \mathbf{1}_{[0,(n+1)/n]}(s) \mathrm{d}s,
$$

with the change of variable $s = (n+1)u$. Assumption (H4) ensures that $\tilde{g}(1/\cdot) \in \mathcal{RV}_{1-\beta}$ so that
$\tilde{g}(s/(n+1))/\tilde{g}(1/(n+1)) \to s^{\beta-1}$ as $n \to +\infty$, for all $s > 0$. Let $\epsilon = \beta/2 > 0$. By Potter bounds [12, Proposition B.1.9], there exists $N \in \mathbb{N} \setminus \{0\}$ such that for all $n \ge N$:

$$
\left| \frac{\tilde{g}(s/(n+1))}{\tilde{g}(1/(n+1))} - 1 \right| \le \max(s^{\beta-1} \min(s^{\epsilon}, s^{-\epsilon})(1-\epsilon) + 1, s^{\beta-1} \max(s^{\epsilon}, s^{-\epsilon})(1+\epsilon) + 1) =: h(s).
$$

Note that $h$ is continuous on $[1,2]$ and thus bounded, while for $s \in (0,1)$, $h(s) = s^{\beta-1-\epsilon}(1+\epsilon) + 1$ is
integrable since $\epsilon = \beta/2 < \beta$. Applying the dominated convergence theorem, it follows that

$$
\int_0^2 \left( \frac{\tilde{g}(s/(n+1))}{\tilde{g}(1/(n+1))} - 1 \right) \mathbf{1}_{[0,(n+1)/n]}(s) \mathrm{d}s \xrightarrow[n \to +\infty]{} \int_0^1 (s^{\beta-1} - 1) \mathrm{d}s = \frac{1}{\beta} - 1 < \infty.
$$

Remark that $\sqrt{n}\tilde{g}(1/(n+1))/(n+1) = \mathcal{O}(n^{1/2-\beta}L(n))$ where $L$ is the slowly-varying function introduced in (H$_4$). Since $\beta > 1/2$, [6, Proposition 1.3.6] yields $\sqrt{n}\tilde{g}(1/(n+1))/(n+1) \to 0$ as $n \to +\infty$. Moreover, since $X_{1,n} \xrightarrow[n \to +\infty]{} x_1 < 0$, it follows $\sqrt{n}\Delta_n \xrightarrow[n \to +\infty]{} 0$. Let us now turn to the remaining terms in $\widehat{D}_{1,g}^{[2]}(\mathcal{X}_n)$. For all $i \ge 2$, one has

$$
\begin{align*}
\left| \int_{\frac{i-1}{n}}^{\frac{i}{n}} \left( \tilde{g}(u) - \tilde{g} \left( \frac{i}{n+1} \right) \right) du \right| &\le \int_{\frac{i-1}{n}}^{\frac{i}{n+1}} \int_{u}^{\frac{i}{n+1}} (-\tilde{g}'(t)) dt du + \int_{\frac{i}{n+1}}^{\frac{i}{n}} \int_{\frac{i}{n+1}}^{u} (-\tilde{g}'(t)) dt du \\
&\le \left( \frac{i}{n+1} - \frac{i-1}{n} \right) \int_{\frac{i-1}{n}}^{\frac{i}{n+1}} (-\tilde{g}'(t)) dt + \left( \frac{i}{n} - \frac{i}{n+1} \right) \int_{\frac{i}{n+1}}^{\frac{i}{n}} (-\tilde{g}'(t)) dt \\
&\le \frac{1}{n+1} \int_{\frac{i-1}{n}}^{\frac{i}{n}} (-\tilde{g}'(t)) dt.
\end{align*}
$$

Recalling that $\tilde{g}(1) = g(0) = 0$, it follows

$$
\begin{align*}
|\Lambda_n| &= \left| \sqrt{n} \sum_{i=2}^{n} X_{i,n} \int_{\frac{i-1}{n}}^{\frac{i}{n}} \left( \tilde{g}(u) - \tilde{g} \left( \frac{i}{n+1} \right) \right) du \right| \\
&\le \frac{\sqrt{n}}{n+1} X_{n,n} \sum_{i=2}^{n} \int_{\frac{i-1}{n}}^{\frac{i}{n}} (-\tilde{g}'(t)) dt \\
&= \frac{\sqrt{n} X_{n,n} \tilde{g}(1/n)}{n+1}.
\end{align*}
$$
Since (C1) holds, usual results on the sample maximum [12, Theorem 1.1.6, Corollary 1.2.4 and Lemma 1.2.9 (3)] yield, as $n \to +\infty$, $X_{n,n} = \mathcal{O}_{\mathbb{P}}(U(n))$ if $x_\infty = +\infty$ and $X_{n,n} = \mathcal{O}_{\mathbb{P}}(1)$ otherwise. Moreover, $\gamma \ge 0$ implies $U \in \mathcal{RV}_\gamma$ in view of [12, Remark 1.2.11]. Then, using (H4) leads to

$$|\Lambda_n| = \begin{cases} \mathcal{O}_{\mathbb{P}}(n^{\gamma+1/2-\beta} L_\infty(n)) \xrightarrow{n \to +\infty} \mathbb{P} \quad 0 & \text{if } x_\infty = +\infty, \\ \mathcal{O}_{\mathbb{P}}(n^{1/2-\beta} L(n)) \xrightarrow{n \to +\infty} \mathbb{P} \quad 0 & \text{otherwise,} \end{cases}$$

since $\beta > 1/2$ and $\gamma < \beta - 1/2$ from Assumption (2) of Theorem 3, where $L_\infty \in \mathcal{RV}_0$. It follows that $\sqrt{n}\widehat{D}_{1,g}^{[2]}(\mathcal{X}_n) \xrightarrow{n \to +\infty} 0$ and consequently

$$Z_{g,n} = \sqrt{n}(\widehat{D}_{1,g}(\mathcal{X}_n) - D_{1,g}(X)) \xrightarrow{n \to +\infty} \mathcal{N}(0, \sigma^2(\tilde{g}, F)).$$

Let us now study $\hat{I}_{1,g}(\mathcal{X}_n)$. As in (B.12), $Z_{1,n} = \sqrt{n}(\hat{\mu}_1 - \mu) \xrightarrow{n \to +\infty} \mathcal{N}(0, \text{Var}(X))$ and calculations similar to those for (B.13) yield as $n \to +\infty$:

$$\sqrt{n}(\hat{I}_{1,g}(\mathcal{X}_n) - I_{1,g}(X)) = \frac{D_{1,g}(X)}{\mu} \left( \frac{Z_{1,n}}{\mu} - \frac{Z_{g,n}}{D_{1,g}(X)} \right) + \mathcal{O}_{\mathbb{P}}\left(\frac{1}{\sqrt{n}}\right),$$

with

$$\begin{aligned} \frac{Z_{1,n}}{\mu} - \frac{Z_{g,n}}{D_{1,g}(X)} &= \sqrt{n} \left( \frac{\hat{\mu}_1 - \mu}{\mu} - \frac{\widehat{D}_{1,g}(\mathcal{X}_n) - D_{1,g}(X)}{D_{1,g}(X)} \right) \\ &= \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^{n} \frac{X_{i,n}}{\mu} - \frac{\widehat{D}_{1,g}^{[1]}(\mathcal{X}_n) + \widehat{D}_{1,g}^{[2]}(\mathcal{X}_n)}{D_{1,g}(X)} \right) \\ &= \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^{n} \left( \frac{X_{i,n}}{\mu} - \frac{\tilde{g}(i/(n+1))X_{i,n}}{D_{1,g}(X)} \right) - \frac{\widehat{D}_{1,g}^{[2]}(\mathcal{X}_n)}{D_{1,g}(X)} \right) \\ &= \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^{n} J\left(\frac{i}{n+1}\right) X_{i,n} \right) - \frac{\sqrt{n}\widehat{D}_{1,g}^{[2]}(\mathcal{X}_n)}{D_{1,g}(X)}. \end{aligned}$$

Similarly, Lemma 8 can be applied to prove that (B.3) and (B.4) also hold for $J$, with the same pair $(r, s)$ as for $g$. Moreover, it is clear that (B.5) in Lemma 9 still holds. Standard results on L-statistics [38, Theorem 1] and [46, Example 1] thus entail that

$$\sqrt{n} \left( \frac{1}{n} \sum_{i=1}^{n} J\left(\frac{i}{n+1}\right) X_{i,n} - \int_{0}^{1} q(u) J(1-u) du \right) = \sqrt{n} \left( \frac{1}{n} \sum_{i=1}^{n} J\left(\frac{i}{n+1}\right) X_{i,n} \right) \xrightarrow{n \to +\infty} \mathcal{N}(0, \sigma^2(J, F)),$$

where $\sigma^2(J, F)$ is defined in (4.3). Finally, combining $\sqrt{n}\widehat{D}_{1,g}^{[2]}(\mathcal{X}_n) \xrightarrow{n \to +\infty} 0$ and $D_{1,g}(X)/\mu = 1 - I_{1,g}(X)$ yields

$$\sqrt{n} \left( \frac{\hat{I}_{1,g}(\mathcal{X}_n) - I_{1,g}(X)}{1 - I_{1,g}(X)} \right) \xrightarrow{n \to +\infty} \mathcal{N}(0, \sigma^2(J, F)).$$

and the result is thus proved. $\square$

*Proof of Example 2.* (i) Simple calculations yield for all $l < 1/\gamma$,

$$\mu_l = \frac{1}{1 - \gamma l}.$$

This result can be applied with $l = 2$, $l = 2\kappa$, $\gamma \in (0, 1/2)$ and $\kappa < 1$.
(ii) One has for all $l > -\theta$,

$$ \mu_l = \frac{\theta}{l + \theta}. $$

This result can be applied with $l = 2$, $l = 2\kappa$ and $\kappa > -\theta/2$.

(iii) Remark that, for all $l > -b$,

$$ \mu_l = \Gamma(1 + 1/b). $$

This result can be applied with $l = 2$, $l = 2\kappa$ and $\kappa > -b/2$.

As a consequence, the moment conditions of Theorem 3 are fulfilled. Finally, it is clear that Assumption (2) of Theorem 3 holds with $(\beta, s) = (1, 2)$ for these three distributions. $\square$

## C Appendix: Tables

<table><caption>Table 1: Usual distributions: Pareto(1, γ) in MDA(Fréchet), Weibull(1, b) in MDA(Gumbel) and Beta(θ, 1), Beta(1, θ) in MDA(Weibull).</caption><thead><tr><th>Distribution</th><th>Support</th><th>Survival function<br>F̄(x)</th><th>Quantile function<br>q(u)</th><th>Expectation<br>μ</th><th>Variance<br>σ²</th></tr></thead><tbody><tr><td>Pareto(1, γ)<br>γ > 0</td><td>[1, +∞)</td><td>x<sup>-1/γ</sup></td><td>u<sup>-γ</sup></td><td><sup>1</sup>&frasl;<sub>1 - γ</sub><br>if γ < 1</td><td><sup>γ²</sup>&frasl;<sub>(1 - γ)<sup>2</sup>(1 - 2γ)</sub><br>if γ < 1/2</td></tr><tr><td>Weibull(1, b)<br>b > 0</td><td>[0, +∞)</td><td>exp(-x<sup>b</sup>)</td><td>log(1/u)<sup>1/b</sup></td><td>Γ(1 + 1/b)</td><td>Γ(1 + 2/b) - Γ(1 + 1/b)<sup>2</sup></td></tr><tr><td>Beta(θ, 1)<br>θ > 0</td><td>[0, 1]</td><td>1 - x<sup>θ</sup></td><td>(1 - u)<sup>1/θ</sup></td><td><sup>θ</sup>&frasl;<sub>θ + 1</sub></td><td><sup>θ</sup>&frasl;<sub>(θ + 2)(θ + 1)<sup>2</sup></sub></td></tr><tr><td>Beta(1, θ)<br>θ > 0</td><td>[0, 1]</td><td>(1 - x)<sup>θ</sup></td><td>1 - u<sup>1/θ</sup></td><td><sup>1</sup>&frasl;<sub>1 + θ</sub></td><td><sup>θ</sup>&frasl;<sub>(θ + 2)(θ + 1)<sup>2</sup></sub></td></tr></tbody></table>
<table><caption>Table 2: Five inequality indices in $(I_{κ,g})$ associated with three usual distributions: Pareto in MDA(Fréchet), Beta in MDA(Weibull) and Weibull in MDA(Gumbel). Recall that $\Gamma$ denotes the Gamma function and the following function is introduced: $(x, y) \in (0, +\infty) \times \mathbb{R} \mapsto \psi(x, y) = -\int_{0}^{+\infty} t^{1/x}e^{-t}k_{y-1}(1 - e^{-t})dt \in [0, +\infty]$, where $k_{y-1}$ is defined in (3.1).</caption><thead><tr><th></th><th>g(u)</th><th>κ</th><th>Pareto(1, γ)<br>γ ∈ (0, 1)</th><th>Beta(θ, 1)<br>θ > 0</th><th>Weibull(1, b)<br>b > 0</th></tr></thead><tbody><tr><td>I<sub>Gini</sub></td><td>2u</td><td>1</td><td>γ/2 - γ</td><td>1/(2θ + 1)</td><td>1 - 2<sup>-1/b</sup></td></tr><tr><td>I<sub>Atkinson, η</sub><br>η ∈ (0, +∞) \ {1}</td><td>1</td><td>1 - η</td><td>1 - (1 - η)[1 - (1 - η)γ]<sup>1/η - 1</sup></td><td>1 - θ + 1/θ (θ/(1 - η + θ))<sup>1/(1 - η)</sup>, if η < 1 + θ<br>1, otherwise</td><td>1 - Γ((1 - η)/b + 1)<sup>1/(1 - η)</sup>, if η < 1 + b<br>1, otherwise</td></tr><tr><td>I<sub>EG, ν</sub><br>ν ≥ 1</td><td>νu<sup>ν-1</sup></td><td>1</td><td>(ν - 1)γ/ν - γ</td><td>1 - ν(θ + 1)B(ν, 1/θ)<br>θ(νθ + 1)</td><td>1 - ν<sup>-1/b</sup></td></tr><tr><td>I<sub>Bonferroni, h</sub><br>h = 1</td><td>-log(1 - u)</td><td>1</td><td>γ[1 + (γ - 1)ψ(1, 1 - γ)]</td><td>1/θ + 1</td><td>1 - ψ(b, 1)/Γ(1 + 1/b)</td></tr><tr><td>I<sub>Mehran</sub></td><td>3u<sup>2</sup></td><td>1</td><td>2γ/3 - γ</td><td>5θ + 1/(2θ + 1)(3θ + 1)</td><td>1 - 3<sup>-1/b</sup></td></tr></tbody></table>