# Zenga's New Index of Economic Inequality, Its Estimation, and an Analysis of Incomes in Italy

Authors: Francesca Greselin; Leo Pasquazzi; Ričardas Zitikis
Journal: Journal of Probability and Statistics
 (2010)

---

## Abstract

For at least a century academics and governmental researchers have been developing measures that would aid them in understanding income distributions, their differences with respect to geographic regions, and changes over time periods. It is a fascinating area due to a number of reasons, one of them being the fact that different measures, or indices, are needed to reveal different features of income distributions. Keeping also in mind that the notions of poor and rich are relative to each other, Zenga (2007) proposed a new index of economic inequality. The index is remarkably insightful and useful, but deriving statistical inferential results has been a challenge. For example, unlike many other indices, Zenga's new index does not fall into the classes ofL-, U-, and V-statistics. In this paper we derive desired statistical inferential results, explore their performance in a simulation study, and then use the results to analyze data from the Bank of Italy Survey on Household Income and Wealth (SHIW).


---

## Body

## 1. Introduction

Measuring and analyzing incomes, losses, risks, and other random outcomes, which we denote byX, has been an active and fruitful research area, particularly in the fields of econometrics and actuarial science. The Gini index is arguably the most popular measure of inequality, with a number of extensions and generalizations available in the literature. Keeping in mind that the notions of poor and rich are relative to each other, Zenga [1] constructed an index that reflects this relativity. We will next recall the definitions of the Gini and Zenga indices.LetF(x)=P[X≤x] denote the cumulative distribution function (cdf) of the random variable X, which we assume to be nonnegative throughout the paper. Let F-1(p)=inf{x:F(x)≥p} denote the corresponding quantile function. The Lorenz curve LF(p) is given by the formula (see [2])(1.1)LF(p)=1μF∫0pF-1(s)ds,
where μF=E[X] is the unknown true mean of X. Certainly, from the rigorous mathematical point of view we should call LF(p) the Lorenz function, but this would deviate from the widely accepted usage of the term “Lorenz curve”. Hence, curves and functions are viewed as synonyms throughout this paper.The classical Gini indexGF can now be written as follows:(1.2)GF=∫01(1-LF(p)p)ψ(p)dp,
where ψ(p)=2p. Note that ψ(p) is a density function on [0,1]. Given the usual econometric interpretation of the Lorenz curve [3], the function(1.3)GF(p)=1-LF(p)p,
which we call the Gini curve, is a relative measure of inequality (see [4]). Indeed, LF(p)/p is the ratio between (i) the mean income of the poorest p×100% of the population and (ii) the mean income of the entire population: the closer to each other these two means are, the lower is the inequality.Zenga's [1] index ZF of inequality is defined by the formula(1.4)ZF=∫01ZF(p)dp,
where the Zenga curve ZF(p) is given by(1.5)ZF(p)=1-LF(p)p·1-p1-LF(p).
The Zenga curve measures the inequality between (i) the poorest p×100% of the population and (ii) the richer remaining (1-p)×100% part of the population by comparing the mean incomes of these two disjoint and exhaustive subpopulations. We will elaborate on this interpretation later, in Section 5.The Gini and Zenga indicesGF and ZF are (weighted) averages of the Gini and Zenga curves GF(p) and ZF(p), respectively. However, while in the case of the Gini index the weight function (i.e., the density) ψ(p)=2p is employed, in the case of the Zenga index the uniform weight function  ψ(p)=1 is used. As a consequence, the Gini index underestimates comparisons between the very poor and the whole population, and emphasizes comparisons which involve almost identical population subgroups. From this point of view, the Zenga index is more impartial: it is based on all comparisons between complementary disjoint population subgroups and gives the same weight to each comparison. Hence, the Zenga index ZF detects, with the same sensibility, all deviations from equality in any part of the distribution.To illustrate the Gini curveGF(p) and its weighted version gF(p)=GF(p)ψ(p), and to also facilitate their comparisons with the Zenga curve ZF(p), we choose the Pareto distribution(1.6)F(x)=1-(x0x)θ,x≥x0,
where x0>0 and θ>0 are parameters. Later in this paper, we will use this distribution in a simulation study, setting x0=1 and θ=2.06. Note that when θ>2, then the second moment of the distribution is finite. The “heavy-tailed” case 1<θ<2 is also of interest, especially when modeling incomes of countries with very high economic inequality. We will provide additional details on the case in Section 5.Note.
Pareto distribution (1.6) is perhaps the oldest model for income distributions. It dates back to Pareto [5], and Pareto [6]. Pareto's original empirical research suggested him that the number of tax payers with income x is roughly proportional to x-(θ+1), where θ is a parameter that measures inequality. For historical details on the interpretation of this parameter in the context of measuring economic inequality, we refer to Zenga [7]. We can view the parameter x0>0 as the lowest taxable income. In addition, besides being the greatest lower bound of the distribution support, x0 is also the scale parameter of the distribution and thus does not affect our inequality indices and curves, as we will see in formulas below.Note.
The Pareto distribution is positively supported,x≥x0>0. In real surveys, however, in addition to many positive incomes we may also observe some zero and negative incomes. This happens when evaluating net household incomes, which are the sums of payroll incomes (net wages, salaries, fringe benefits), pensions and net transfers (pensions, arrears, financial assistance, scholarships, alimony, gifts). Paid alimony and gifts are subtracted in forming the incomes. However, negative incomes usually happen in the case of very few statistical units. For example, in the 2006 Bank of Italy survey we observe only four households with nonpositive incomes, out of the total of 7,766 households. Hence, it is natural to fit the Pareto model to the positive incomes and keep in mind that we are actually dealing with a conditional distribution. If, however, it is desired to deal with negative, null, and positive incomes, then instead of the Pareto distribution we may switch to different ones, such as Dagum distributions with three or four parameters [8–10].Corresponding to Pareto distribution (1.6), the Lorenz curve is given by the formula LF(p)=1-(1-p)1-1/θ (see [11]), and thus the Gini curve becomes GF(p)=((1-p)1-1/θ-(1-p))/p. In Figure 1(a) we have depicted the Gini and weighted Gini curves. The corresponding Zenga curve is equal to ZF(p)=(1-(1-p)1/θ)/p and is depicted in Figure 1(b), alongside the Gini curve GF(p) for an easy comparison. Figure 1(a) allows us to appreciate how the Gini weight function ψ(p)=2p disguises the high inequality between the mean income of the very poor and that of the whole population, and overemphasizes comparisons between almost identical subgroups. The outcome is that the Gini index GF underestimates inequality. In Figure 1(b) we see the difference between the Gini and Zenga inequality curves. For example, GF(p) for p=0.8 yields 0.296, which tells us that the mean income of the poorest 80% of the population is 29.6% lower than the mean income of the whole population, while the corresponding ordinate of the Zenga curve is ZF(0.8)=0.678, which tells us that the mean income of the poorest 80% of the population is 67.8% lower than the mean income of the remaining (richer) part of the population.The Gini curveGF(p)(dashed; (a) and (b)), the weighted Gini curve gF(p)(solid; (a)), and the Zenga curve ZF(p)(solid; (b)) in the Pareto case with x0=1 and θ=2.06.
(a)(b)The rest of this paper is organized as follows. In Section2 we define two estimators of the Zenga index ZF and develop statistical inferential results. In Section 3 we present results of a simulation study, which explores the empirical performance of two Zenga estimators, Ẑn and Z̃n, including coverage accuracy and length of several types of confidence intervals. In Section 4 we present an analysis of the the Bank of Italy Survey on Household Income and Wealth (SHIW) data. In Section 5 we further contribute to the understanding of the Zenga index ZF by relating it to lower and upper conditional expectations, as well as to the conditional tail expectation (CTE), which has been widely used in insurance. In Section 6 we provide a theoretical background of the aforementioned two empirical Zenga estimators. In Section 7 we justify the definitions of several variance estimators as well as their uses in constructing confidence intervals. In Section 8 we prove Theorem 2.1 of Section 2, which is the main technical result of the present paper. Technical lemmas and their proofs are relegated to Section 9.


## 2. Estimators and Statistical Inference

Unless explicitly stated otherwise, our statistical inferential results are derived under the assumption that data are outcomes of independent and identically distributed (i.i.d.) random variables.Hence, letX1,…,Xn be independent copies of X. We use two nonparametric estimators for the Zenga index ZF. The first one [12] is given by the formula(2.1)Ẑn=1-1n∑i=1n-1i-1∑k=1iXk:n(n-i)-1∑k=i+1nXk:n,
where X1:n≤⋯≤Xn:n are the order statistics of X1,…,Xn. With X̅ denoting the sample mean of X1,…,Xn, the second estimator of the Zenga index ZF is given by the formula(2.2)Z̃n=-∑i=2n∑k=1i-1Xk:n-(i-1)Xi:n∑k=i+1nXk:n+iXi:nlog(ii-1)+∑i=1n-1(X̅Xi:n-1-∑k=1i-1Xk:n-(i-1)Xi:n∑k=i+1nXk:n+iXi:n)log(1+Xi:n∑k=i+1nXk:n).
The two estimators Ẑn and Z̃n are asymptotically equivalent. However, despite the fact that the estimator Z̃n is more complex, it will nevertheless be more convenient to work with when establishing asymptotic results later in this paper.Unless explicitly stated otherwise, we assume throughout that the cdfF(x) of X is a continuous function. We note that continuous cdf's are natural choices when modeling income distributions, insurance risks, and losses (see, e.g., [13]).Theorem 2.1.
If the momentE[X2+α] is finite for some α>0, then one has the asymptotic representation
(2.3)n(Z̃n-ZF)=1n∑i=1nh(Xi)+oP(1),
where oP(1) denotes a random variable that converges to 0 in probability when n→∞, and
(2.4)h(Xi)=∫0∞(1{Xi≤x}-F(x))wF(F(x))dx
with the weight function
(2.5)wF(t)=-1μF∫0t(1p-1)LF(p)(1-LF(p))2dp+1μF∫t1(1p-1)11-LF(p)dp.In view of Theorem2.1, the asymptotic distribution of n(Z̃n-ZF) is centered normal with the variance σF2=E[h2(X)], which is finite (see Theorem 7.1) and can be written as follows:(2.6)σF2=∫0∞∫0∞(min{F(x),F(y)}-F(x)F(y))wF(F(x))wF(F(y))dxdy.
Alternatively,(2.7)σF2=∫01(∫[0,u)twF(t)dF-1(t)-∫[u,1)(1-t)wF(t)dF-1(t))2du.
The latter expression of σF2 is particularly convenient when working with distributions for which the first derivative (when it exists) of the quantile F-1(t) is a relatively simple function, as is the case for a large class of distributions (see, e.g., [14]). However, irrespectively of what expression for the variance σF2 we use, the variance is unknown since the cdf F(x) is unknown, and thus σF2 needs to be estimated empirically.


### 2.1. One Sample Case

Replacing the population cdf everywhere on the right-hand side of (2.6) by the empirical cdf Fn(x)=n-1∑i=1n1{Xi≤x}, where 1 denotes the indicator function, we obtain (Theorem 7.2) the following estimator of the variance σF2:(2.8)SX,n2=∑k=1n-1∑l=1n-1(min{k,l}n-knln)×wX,n(kn)wX,n(ln)(Xk+1:n-Xk:n)(Xl+1:n-Xl:n),
where(2.9)wX,n(kn)=-∑i=1kIX,n(i)+∑i=k+1nJX,n(i)
with the following expressions for the summands IX,n(i) and JX,n(i): first,(2.10)IX,n(1)=-∑k=2nXk:n-(n-1)X1:n(∑k=1nXk:n)(∑k=2nXk:n)+1X1,nlog(1+X1:n∑k=2nXk:n).
Furthermore, for every i=2,…,n-1,(2.11)IX,n(i)=n∑k=1i-1Xk:n-(i-1)Xi:n(∑k=i+1nXk:n+iXi:n)2log(ii-1)-(∑k=i+1nXk:n-(n-i)Xi:n)(∑k=1nXk:n)(∑k=i+1nXk:n+iXi:n)(∑k=i+1nXk:n)(∑k=inXk:n)+(1Xi:n+n∑k=1i-1Xk:n-(i-1)Xi:n(∑k=i+1nXk:n+iXi:n)2)log(1+Xi:n∑k=i+1nXk:n),(2.12)JX,n(i)=n∑k=i+1nXk:n+iXi:nlog(ii-1)-∑k=i+1nXk:n-(n-i)Xi:nXi:n(∑k=i+1nXk:n+iXi:n)log(1+Xi:n∑k=i+1nXk:n).
Finally,(2.13)JX,n(n)=1Xn,nlog(nn-1).
With the just defined estimator SX,n2 of the variance σF2, we have the asymptotic result:(2.14)n(Z̃n-ZF)SX,n→d𝒩(0,1),
where →d denotes convergence in distribution.


### 2.2. Two Independent Samples

We now discuss a variant of statement (2.14) in the case of two populations when samples are independent. Namely, let the random variables X1,…,Xn~F and Y1,…,Ym~H be independent within and between the two samples. Just like in the case of the cdf F(x), here we also assume that the cdf H(x) is continuous and E[Y2+α]<∞ for some α>0. Furthermore, we assume that the sample sizes n and m are comparable, which means that there exists η∈(0,1) such that(2.15)mn+m→η∈(0,1)
when both n and m tend to infinity. From statement (2.3) and its counterpart for Yi~H we then have that the quantity nm/(n+m)((Z̃X,n-Z̃Y,m)-(ZF-ZH)) is asymptotically normal with mean zero and the variance ησF2+(1-η)σH2. To estimate the variances σF2 and σH2, we use SX,n2 and SY,n2, respectively, and obtain the following result:(2.16)(Z̃X,n-Z̃Y,m)-(ZF-ZH)(1/n)SX,n2+(1/m)SY,m2→d𝒩(0,1).


### 2.3. Paired Samples

Consider now the case when the two samplesX1,…,Xn~F and Y1,…,Ym~H are paired. Thus, we have that m=n, and we also have that the pairs (X1,Y1),…,(Xn,Yn) are independent and identically distributed. Nothing is assumed about the joint distribution of (X,Y). As before, the cdf's F(x) and H(y) are continuous and both have finite moments of order 2+α, for some α>0. From statement (2.3) and its analog for Y we have that n((Z̃X,n-Z̃Y,n)-(ZF-ZH)) is asymptotically normal with mean zero and the variance σF,H2=E[(h(X)-h(Y))2]. The latter variance can of course be written as σF2-2E[h(X)h(Y)]+σH2. Having already constructed estimators SX,n2 and SY,n2, we are only left to construct an estimator for E[h(X)h(Y)]. (Note that when X and Y are independent, then P[X≤x,Y≤y]=F(x)H(y) and thus the expectation E[h(X)h(Y)] vanishes.) To this end, we write the equation(2.17)E[h(X)h(Y)]=∫0∞∫0∞(P[X≤x,Y≤y]-F(x)H(y))wF(F(x))wH(H(y))dxdy.
Replacing the cdf's F(x) and H(y) everywhere on the right-hand side of the above equation by their respective empirical estimators Fn(x) and Hn(y), we have (Theorem 7.3)(2.18)SX,Y,n=∑k=1n-1∑l=1n-1(1n∑i=1k1{Y(i,n)≤Yl:n}-knln)×wX,n(kn)wY,n(ln)(Xk+1:n-Xk:n)(Yl+1:n-Yl:n),
where Y(1,n),…,Y(n,n) are the induced (by X1,…,Xn) order statistics of Y1,…,Yn. (Note that when Y≡X, then Y(i,n)=Yi:n and so the sum ∑i=1k1{Y(i,n)≤Yl:n} is equal to min{k,l}; hence, estimator (2.18) coincides with estimator (2.8), as expected.) Consequently, SX,n2-2SX,Y,n+SY,n2 is an empirical estimator of σF,H2, and so we have that(2.19)n(Z̃X,n-Z̃Y,n)-(ZF-ZH)SX,n2-2SX,Y,n+SY,n2→d𝒩(0,1).We conclude this section with a note that the above established asymptotic results (2.14), (2.16), and (2.19) are what we typically need when dealing with two populations, or two time periods, but extensions to more populations and/or time periods would be a worthwhile contribution. For hints and references on the topic, we refer to Jones et al. [15] and Brazauskas et al. [16].


## 2.1. One Sample Case

Replacing the population cdf everywhere on the right-hand side of (2.6) by the empirical cdf Fn(x)=n-1∑i=1n1{Xi≤x}, where 1 denotes the indicator function, we obtain (Theorem 7.2) the following estimator of the variance σF2:(2.8)SX,n2=∑k=1n-1∑l=1n-1(min{k,l}n-knln)×wX,n(kn)wX,n(ln)(Xk+1:n-Xk:n)(Xl+1:n-Xl:n),
where(2.9)wX,n(kn)=-∑i=1kIX,n(i)+∑i=k+1nJX,n(i)
with the following expressions for the summands IX,n(i) and JX,n(i): first,(2.10)IX,n(1)=-∑k=2nXk:n-(n-1)X1:n(∑k=1nXk:n)(∑k=2nXk:n)+1X1,nlog(1+X1:n∑k=2nXk:n).
Furthermore, for every i=2,…,n-1,(2.11)IX,n(i)=n∑k=1i-1Xk:n-(i-1)Xi:n(∑k=i+1nXk:n+iXi:n)2log(ii-1)-(∑k=i+1nXk:n-(n-i)Xi:n)(∑k=1nXk:n)(∑k=i+1nXk:n+iXi:n)(∑k=i+1nXk:n)(∑k=inXk:n)+(1Xi:n+n∑k=1i-1Xk:n-(i-1)Xi:n(∑k=i+1nXk:n+iXi:n)2)log(1+Xi:n∑k=i+1nXk:n),(2.12)JX,n(i)=n∑k=i+1nXk:n+iXi:nlog(ii-1)-∑k=i+1nXk:n-(n-i)Xi:nXi:n(∑k=i+1nXk:n+iXi:n)log(1+Xi:n∑k=i+1nXk:n).
Finally,(2.13)JX,n(n)=1Xn,nlog(nn-1).
With the just defined estimator SX,n2 of the variance σF2, we have the asymptotic result:(2.14)n(Z̃n-ZF)SX,n→d𝒩(0,1),
where →d denotes convergence in distribution.


## 2.2. Two Independent Samples

We now discuss a variant of statement (2.14) in the case of two populations when samples are independent. Namely, let the random variables X1,…,Xn~F and Y1,…,Ym~H be independent within and between the two samples. Just like in the case of the cdf F(x), here we also assume that the cdf H(x) is continuous and E[Y2+α]<∞ for some α>0. Furthermore, we assume that the sample sizes n and m are comparable, which means that there exists η∈(0,1) such that(2.15)mn+m→η∈(0,1)
when both n and m tend to infinity. From statement (2.3) and its counterpart for Yi~H we then have that the quantity nm/(n+m)((Z̃X,n-Z̃Y,m)-(ZF-ZH)) is asymptotically normal with mean zero and the variance ησF2+(1-η)σH2. To estimate the variances σF2 and σH2, we use SX,n2 and SY,n2, respectively, and obtain the following result:(2.16)(Z̃X,n-Z̃Y,m)-(ZF-ZH)(1/n)SX,n2+(1/m)SY,m2→d𝒩(0,1).


## 2.3. Paired Samples

Consider now the case when the two samplesX1,…,Xn~F and Y1,…,Ym~H are paired. Thus, we have that m=n, and we also have that the pairs (X1,Y1),…,(Xn,Yn) are independent and identically distributed. Nothing is assumed about the joint distribution of (X,Y). As before, the cdf's F(x) and H(y) are continuous and both have finite moments of order 2+α, for some α>0. From statement (2.3) and its analog for Y we have that n((Z̃X,n-Z̃Y,n)-(ZF-ZH)) is asymptotically normal with mean zero and the variance σF,H2=E[(h(X)-h(Y))2]. The latter variance can of course be written as σF2-2E[h(X)h(Y)]+σH2. Having already constructed estimators SX,n2 and SY,n2, we are only left to construct an estimator for E[h(X)h(Y)]. (Note that when X and Y are independent, then P[X≤x,Y≤y]=F(x)H(y) and thus the expectation E[h(X)h(Y)] vanishes.) To this end, we write the equation(2.17)E[h(X)h(Y)]=∫0∞∫0∞(P[X≤x,Y≤y]-F(x)H(y))wF(F(x))wH(H(y))dxdy.
Replacing the cdf's F(x) and H(y) everywhere on the right-hand side of the above equation by their respective empirical estimators Fn(x) and Hn(y), we have (Theorem 7.3)(2.18)SX,Y,n=∑k=1n-1∑l=1n-1(1n∑i=1k1{Y(i,n)≤Yl:n}-knln)×wX,n(kn)wY,n(ln)(Xk+1:n-Xk:n)(Yl+1:n-Yl:n),
where Y(1,n),…,Y(n,n) are the induced (by X1,…,Xn) order statistics of Y1,…,Yn. (Note that when Y≡X, then Y(i,n)=Yi:n and so the sum ∑i=1k1{Y(i,n)≤Yl:n} is equal to min{k,l}; hence, estimator (2.18) coincides with estimator (2.8), as expected.) Consequently, SX,n2-2SX,Y,n+SY,n2 is an empirical estimator of σF,H2, and so we have that(2.19)n(Z̃X,n-Z̃Y,n)-(ZF-ZH)SX,n2-2SX,Y,n+SY,n2→d𝒩(0,1).We conclude this section with a note that the above established asymptotic results (2.14), (2.16), and (2.19) are what we typically need when dealing with two populations, or two time periods, but extensions to more populations and/or time periods would be a worthwhile contribution. For hints and references on the topic, we refer to Jones et al. [15] and Brazauskas et al. [16].


## 3. A Simulation Study

Here we investigate the numerical performance of the estimatorsẐn and Z̃n by simulating data from Pareto distribution (1.6) with x0=1 and θ=2.06. These choices give the value ZF=0.6, which is approximately seen in real income distributions. As to the (artificial) choice x0=1, we note that since x0 is the scale parameter in the Pareto model, the inequality indices and curves are invariant to it. Hence, all results to be reported in this section concerning the coverage accuracy and size of confidence intervals will not be affected by the choice x0=1.Following Davison and Hinkley [17, Chapter 5], we compute four types of confidence intervals: normal, percentile, BCa, and t-bootstrap. For normal and studentized bootstrap confidence intervals we estimate the variance using empirical influence values. For the estimator Z̃n, the influence values h(Xi) are obtained from Theorem 2.1, and those for the estimator Ẑn using numerical differentiation as in Greselin and Pasquazzi [12].In Table1  we report coverage percentages of 10,000 confidence intervals, for each of the four types: normal, percentile, BCa, and t-bootstrap. Bootstrap-based approximations have been obtained from 9,999 resamples of the original samples. As suggested by Efron [18], we have approximated the acceleration constant for the BCa confidence intervals by one-sixth times the standardized third moment of the influence values. In Table 2 we report summary statistics concerning the size of the 10,000 confidence intervals. As expected, the confidence intervals based on Ẑn and Z̃n exhibit similar characteristics. We observe from Table 1 that all confidence intervals suffer from some undercoverage. For example, with sample size 800, about 97.5% of the studentized bootstrap confidence intervals with 0.99 nominal confidence level contain the true value of the Zenga index. It should be noted that the higher coverage accuracy of the studentized bootstrap confidence intervals (when compared to the other ones) comes at the cost of their larger sizes, as seen in Table 2.  Some of the studentized bootstrap confidence intervals extend beyond the range [0,1] of the Zenga index ZF, but this can easily be fixed by taking the minimum between the currently recorded upper bounds and 1, which is the upper bound of the Zenga index ZF for every cdf F. We note that for the BCa confidence intervals, the number of bootstrap replications of the original sample has to be increased beyond 9,999 if the nominal confidence level is high. Indeed, for samples of size 800, it turns out that the upper bound of 1,598 (out of 10,000) of the BCa confidence intervals based on Ẑn and with 0.99 nominal confidence level is given by the largest order statistics of the bootstrap distribution. For the confidence intervals based on Z̃n, the corresponding figure is 1,641.Table 1
Coverage proportions of confidence intervals from the Pareto parent distribution withx0=1 and θ=2.06 (ZF=0.6).
ẐnZ̃n0.90000.95000.97500.99000.90000.95000.97500.9900nNormal confidence intervals2000.79150.85600.89540.92810.78810.85270.89260.92664000.80590.87050.90830.94090.80470.86930.90780.93968000.82560.88890.92450.95140.82460.88820.92370.9503nPercentile confidence intervals2000.77630.83260.86840.90020.76290.81900.85670.88924000.80040.85430.89190.92180.79340.84870.88640.91798000.82100.87770.91380.94150.81680.87510.91190.9393nBCa confidence intervals2000.80820.86840.90770.93830.80540.8670.90470.93744000.82050.88630.92260.95310.82040.8860.92120.95238000.83430.89870.93310.96340.83380.89830.93230.9634nt-bootstrap confidence intervals2000.84750.90410.93850.96580.84850.90490.94000.96754000.85350.91240.94620.97080.85340.91200.94630.97098000.85800.91680.95070.97580.85720.91690.95040.9754Table 2
Size of the 95% asymptotic confidence intervals from the Pareto parent distribution withx0=1 and θ=2.06 (ZF=0.6).
ẐnZ̃nminmeanmaxminmeanmaxnNormal confidence intervals2000.06800.14930.72630.06740.15000.73004000.05640.11640.74460.05630.11670.74658000.04620.08990.65280.04620.09000.6535nPercentile confidence intervals2000.06730.14560.47510.06670.14620.47824000.05610.11400.47120.05610.11430.47218000.04670.08830.41100.04680.08840.4117nBCa confidence intervals2000.06680.14910.46320.06610.14970.46524000.05610.11830.46250.05580.11860.46298000.04650.09250.40830.04670.09270.4085nt-bootstrap confidence intervals2000.06770.20682.43070.06800.20992.51484000.05720.15502.08510.05730.15592.10098000.04730.11592.20150.04740.11622.2051


## 4. An Analysis of Italian Income Data

In this section we use the Zenga indexZF to analyze data from the Bank of Italy Survey on Household Income and Wealth (SHIW). The sample of the 2006 wave of this survey contains 7,768 households, with 3,957 of them being panel households. For detailed information on the survey, we refer to the Bank of Italy [19] publication. In order to treat data correctly in the case of different household sizes, we work with equivalent incomes, which we have obtained by dividing the total household income by an equivalence coefficient, which is the sum of weights assigned to each household member. Following the modified Organization for Economic Cooperation and Development (OECD) equivalence scale, we give weight 1 to the household head, 0.5 to the other adult members of the household, and 0.3 to the members under 14 years of age.  It should be noted, however, that—as is the case in many surveys concerning income analysis—households are selected using complex sampling designs. In such cases, statistical inferential results are quite complex. To alleviate the difficulties, in the present paper we follow the commonly accepted practice and treat income data as if they were i.i.d.In Table3 we report the values of Ẑn and Z̃n according to the geographic area of the households, and we also report confidence intervals for ZF based on the two estimators. We note that two households in the sample had negative incomes in 2006, and so we have not included them in our computations.Table 3
Confidence intervals forZF in the 2006 Italian income distribution.
Ẑn estimatorZ̃n estimator95%99%95%99%LowerUpperLowerUpperLowerUpperLowerUpperNorthwest:n=1988, Ẑn=0.5953, Z̃n=0.5948Normal0.57750.61440.57170.62020.57710.61380.57130.6196Student0.57860.61680.57370.62400.57910.61720.57480.6243Percent0.57630.61320.57100.61930.57580.61240.57060.6185BCa0.57890.61600.57410.62340.57850.61560.57380.6226Northeast:n=1723, Ẑn=0.6108, Z̃n=0.6108Normal0.58490.63930.57640.64780.58490.63930.57640.6479Student0.58740.65260.57960.66690.58970.65380.58360.6685Percent0.58400.63790.57730.64760.58390.63790.57720.6475BCa0.58940.64780.58410.66160.58940.64790.58420.6615Center:n=1574, Ẑn=0.6316, Z̃n=0.6316Normal0.59570.67080.58390.68260.59560.67080.58380.6827Student0.59910.69910.58970.72840.60360.70160.59770.7311Percent0.59480.66890.58640.68180.59480.66880.58630.6818BCa0.60240.68500.59630.70210.60240.68500.59630.7020South:n=1620, Ẑn=0.6557, Z̃n=0.6543Normal0.63580.67700.62930.68340.63460.67560.62820.6820Student0.63710.68050.63130.69020.63710.67960.63200.6900Percent0.63510.67570.62860.68280.63370.67420.62760.6812BCa0.63750.67930.63250.68880.63630.67780.63150.6873Islands:n=861, Ẑn=0.6109, Z̃n=0.6095Normal0.59180.63170.58560.63800.59100.63020.58480.6364Student0.59270.63390.58640.64050.59280.63300.58740.6401Percent0.58970.62970.58390.63600.58850.62750.58310.6340BCa0.59230.63240.58680.64140.59140.63070.58600.6394Italy (entire population):n=7766, Ẑn=0.6470, Z̃n=0.6464Normal0.63460.65960.63070.66360.63410.65910.63020.6630Student0.63590.66290.63270.66860.63580.66270.63310.6683Percent0.63480.65970.63140.66400.63430.65920.63090.6635BCa0.63630.66190.63340.66760.63580.66130.63300.6669Note.
Removing the negative incomes from our current analysis is important as otherwise we would need to develop a much more complex methodology than the one offered in this paper. To give a flavour of technical challenges, we note that the Gini index may overestimate the economic inequality when negative, zero, and positive incomes are considered. In this case the Gini index needs to be renormalized as demonstrated by, for example, Chen et al. [20]. Another way to deal with the issue would be to analyze the negative incomes and their concentration separately from the zero and positive incomes and their concentration.Consequently, the point estimates ofZF are based on 7,766 equivalent incomes with Ẑn=0.6470 and Z̃n=0.6464. As pointed out by Maasoumi [21], however, good care is needed when comparing point estimates of inequality measures. Indeed, direct comparison of the point estimates corresponding to the five geographic areas of Italy would lead us to the conclusion that the inequality is higher in the central and southern areas when compared to the northern area and the islands. But as we glean from pairwise comparisons of the confidence intervals, only the differences between the estimates corresponding to the northwestern and southern areas and perhaps to the islands and the southern area may be deemed statistically significant.Moreover, we have used the paired samples of the 2004 and 2006 incomes of the 3,957 panel households in order to check whether during this time period there was a change in inequality among households. In Table4  we report the values of Z̃n based on the panel households for these two years, and the 95% confidence intervals for the difference between the values of the Zenga index for the years 2006 and 2004. These computations have been based on formula (2.19). Having removed the four households with at least one negative income in the paired sample, we were left with a total of 3,953 observations. We see that even though we deal with large sample sizes, the point estimates alone are not reliable. Indeed, for Italy as the whole and for all geographic areas except the center, the point estimates suggest that the Zenga index decreased from the year 2004 to 2006. However, the 95% confidence intervals in Table 4 suggest that this change is not significant.Table 4
95% confidence intervals for the difference of the Zenga indices between 2006 and 2004 in the Italian income distribution.
Northwest (926 pairs)Northeast (841 pairs)Center (831 pairs)Z̃n(2006)0.5797Z̃n(2006)0.6199Z̃n(2006)0.5921Z̃n(2004)0.5955Z̃n(2004)0.6474Z̃n(2004)0.5766Difference-0.0158Difference-0.0275Difference0.0155LowerUpperLowerUpperLowerUpperNormal-0.04260.0102-0.05730.0003-0.01830.0514Student-0.04630.0103-0.05910.0017-0.01560.0644Percent-0.04210.0108-0.05370.0040-0.01830.0505BCa-0.04400.0087-0.05510.0022-0.01300.0593South (843 pairs)Islands (512 pairs)Italy (3953 pairs)Z̃n(2006)0.6200Z̃n(2006)0.6179Z̃n(2006)0.6362Z̃n(2004)0.6325Z̃n(2004)0.6239Z̃n(2004)0.6485Difference-0.0125Difference-0.0060Difference-0.0123LowerUpperLowerUpperLowerUpperNormal-0.03720.0129-0.03330.0213-0.02590.0007Student-0.03650.0166-0.03510.0222-0.02640.0013Percent-0.03720.0131-0.03330.0214-0.02530.0016BCa-0.03510.0162-0.03310.0216-0.02550.0013


## 5. An Alternative Look at the Zenga Index

In various contexts we have notions of rich and poor, large and small, risky and secure. They divide the underlying populations into two parts, which we view as subpopulations. The quantileF-1(p), for some p∈(0,1), usually serves as a boundary separating the two subpopulations. For example, we may define rich if X>F-1(p) and poor if X≤F-1(p). Calculating the mean value of the former subpopulation gives rise to the upper conditional expectation E[X∣X>F-1(p)], which is known in the actuarial risk theory as the conditional tail expectation (CTE). Calculating the mean value of the latter subpopulation gives rise to the lower conditional expectation E[X∣X≤F-1(p)], which is known in the econometric literature as the absolute Bonferroni curve, as a function of p.Clearly, the ratio(5.1)RF(p)=E[X∣X≤F-1(p)]E[X∣X>F-1(p)]
of the lower and upper conditional expectations takes on values in the interval [0,1]. When X is equal to any constant, which can be interpreted as the egalitarian case, then RF(p) is equal to 1. The ratio RF(p) is equal to 0 for all p∈(0,1) when the lower conditional expectation is equal to 0 for all p∈(0,1). This means extreme inequality in the sense that, loosely speaking, there is only one individual who possesses the entire wealth. Our wish to associate the egalitarian case with 0 and the extreme inequality with 1 leads to function 1-RF(p), which coincides with the Zenga curve (1.5) when the cdf F(x) is continuous. The area(5.2)1-∫01E[X∣X≤F-1(p)]E[X∣X>F-1(p)]dp
beneath the function 1-RF(p) is always in the interval [0,1]. Quantity (5.2) is a measure of inequality and coincides with the earlier defined Zenga index ZF when the cdf F(x) is continuous, which we assume throughout the paper.Note that under the continuity ofF(x), the lower and upper conditional expectations are equal to the absolute Bonferroni curve p-1ALF(p) and the dual absolute Bonferroni curve (1-p)-1(μF-ALF(p)), respectively, where(5.3)ALF(p)=∫0pF-1(t)dt
is the absolute Lorenz curve. This leads us to the expression of the Zenga index ZF given by (1.4), which we now rewrite in terms of the absolute Lorenz curve as follows:(5.4)ZF=1-∫01(1p-1)ALF(p)μF-ALF(p)dp.
We will extensively use expression (5.4) in the proofs below. In particular, we will see in the next section that the empirical Zenga index Z̃n is equal to ZF with the population cdf F(x) replaced by the empirical cdf Fn(x).We are now in the position to provide additional details on the earlier noted Pareto case1<θ<2, when the Pareto distribution has finite E[X] but infinite E[X2]. The above derived asymptotic results and thus the statistical inferential theory fail in this case. The required adjustments are serious and rely on the use of the extreme value theory, instead of the classical central limit theorem (CLT). Specifically, the task can be achieved by first expressing the absolute Lorenz curve ALF(p) in terms of the conditional tail expectation (CTE):(5.5)CTEF(p)=11-p∫p1F-1(t)dt
using the equation ALF(p)=μF-(1-p)CTEF(p). Hence, (5.4) becomes(5.6)ZF=1-∫011p(CTEF(0)CTEF(p)-(1-p))dp,
where CTEF(0) is of course the mean μF. Note that replacing the population cdf F(x) by its empirical counterpart Fn(x) on the right-hand side of (5.6) would not lead to an estimator that would work when E[X2]=∞, and thus when the Pareto parameter 1<θ<2. A solution to this problem is provided by Necir et al. [22], who have suggested a new estimator of the conditional tail expectation CTEF(p) for heavy-tailed distributions. Plugging in that estimator instead of the CTE on the right-hand side of (5.6) produces an estimator of the Zenga index when E[X2]=∞. Establishing asymptotic results for the new “heavy-tailed” Zenga estimator would, however, be a complex technical task, well beyond the scope of the present paper, as can be seen from the proofs of Necir et al. [22].


## 6. A Closer Look at the Two Zenga Estimators

Since samples are “discrete populations”, (5.2) and (5.4) lead to slightly different empirical estimators of ZF. If we choose (5.2) and replace all population-related quantities by their empirical counterparts, then we will arrive at the estimator Ẑn, as seen from the proof of the following theorem.Theorem 6.1.
The empirical Zenga indexẐn is an empirical estimator of ZF.Proof.
LetU be a uniform on [0,1] random variable independent of X. The cdf of F-1(U) is F. Hence, we have the following equations:
(6.1)ZF=1-EU(EX[X∣X≤F-1(U)]EX[X∣X>F-1(U)])=1-∫(0,∞)1-F(x)F(x)E[X1{X≤x}]E[X1{X>x}]dF(x)=1-∫(0,∞)1-F(x)F(x)∫(0,x]ydF(y)∫(x,∞)ydF(y)dF(x).
Replacing every F on the right-hand side of (6.1) by Fn, we obtain
(6.2)1-1n∑i=1n-11-Fn(Xi:n)Fn(Xi:n)∑k=1nXk:n1{Xk:n≤Xi:n}∑k=1nXk:n1{Xk:n>Xi:n},
which simplifies to
(6.3)1-1n∑i=1n-11-i/ni/n∑k=1iXk:n∑k=i+1nXk:n.
This is the estimator Ẑn [12].If, on the other hand, we choose (5.4) as the starting point for constructing an empirical estimator for ZF, then we first replace the quantile F-1(p) by its empirical counterpart(6.4)Fn-1(p)=inf{x:Fn(x)≥p}=Xi:nwhenp∈((i-1)n,in]
in the definition of ALF(p), which leads to the empirical absolute Lorenz curve ALn(p), and then we replace each ALF(p) on the right-hand side of (5.4) by the just constructed ALn(p). (Note that μF=ALF(1)≈ALn(1)=X¯.) These considerations produce the empirical Zenga index Z̃n, as seen from the proof of the following theorem.Theorem 6.2.
The empirical Zenga indexZ̃n is an estimator of ZF.Proof.
By construction, the estimatorZ̃n is given by the equation:
(6.5)Z̃n=1-∫01(1p-1)ALn(p)X̅-ALn(p)dp.
Hence, the proof of the lemma reduces to verifying that the right-hand sides of (2.2) and (6.5) coincide. For this, we split the integral in (6.5) into the sum of integrals over the intervals ((i-1),i/n) for i=1,…,n. For every p∈((i-1)/n,i/n), we have ALn(p)=Ci,n+pXi:n, where
(6.6)Ci,n=1n∑k=1i-1Xk:n-i-1nXi:n.
Hence, (6.5) can be rewritten as Z̃n=∑i=1nζi,n, where
(6.7)ζi,n=1n-∫(i-1)/ni/n(1p-1)Λi,n+pΨi,n-pdp
with
(6.8)Λi,n=Ci,nXi:n,Ψi,n=X̅-Ci,nXi:n.
Consider first the case i=1. We have C1,n=0 and thus Λ1,n=0, which implies
(6.9)ζ1,n=(X̅X1:n-1)log(1+X1:n∑k=2nXk:n).
Next, we consider the case i=n. We have Cn,n=X̅-Xn:n and thus Ψn,n=1, which implies
(6.10)ζn,n=(1-X̅Xn:n)log(nn-1).
When 2≤i≤n-1, then the integrand in the definition of ζi,n does not have any singularity, since Ψi,n>i/n due to ∑k=i+1nXk:n>0 almost surely. Hence, after simple integration we have that, for i=2,…,n-1,
(6.11)ζi,n=(i-1)Xi:n-∑k=1i-1Xk:n∑k=i+1nXk:n+iXi:nlog(ii-1)+(X̅Xi:n-1+(i-1)Xi:n-∑k=1i-1Xk:n∑k=i+1nXk:n+iXi:n)log(1+Xi:n∑k=i+1nXk:n).
With the above formulas for ζi,n we easily check that the sum ∑i=1nζi,n is equal to the right-hand side of (2.2). This completes the proof of Theorem 6.2.


## 7. A Closer Look at the Variances

Following the formulation of Theorem2.1 we claimed that the asymptotic distribution of n(Z̃n-ZF) is centered normal with the finite variance σF2=E[h2(X)]. The following theorem proves this claim.Theorem 7.1.
WhenE[X2+α]<∞ for some α>0, then n-1/2∑i=1nh(Xi) converges in distribution to the centered normal random variable
(7.1)Γ=∫0∞ℬ(F(x))wF(F(x))dx,
where ℬ(p) is the Brownian bridge on the interval [0,1]. The variance of Γ is finite and equal to σF2.Proof.
Note thatn-1/2∑i=1nh(Xi) can be written as ∫0∞en(F(x))wF(F(x))dx, where en(p)=n(En(p)-p) is the empirical process based on the uniform on [0,1] random variables Ui=F(Xi),i=1,…,n. We will next show that
(7.2)∫0∞en(F(x))wF(F(x))dx→d∫0∞ℬ(F(x))wF(F(x))dx.
The proof is based on the well-known fact that, for every ɛ>0, the following weak convergence of stochastic processes takes place:
(7.3){en(p)p1/2-ɛ(1-p)1/2-ɛ,0≤p≤1}⇒{ℬ(p)p1/2-ɛ(1-p)1/2-ɛ,0≤p≤1}.
Hence, in order to prove statement (7.2), we only need to check that the integral
(7.4)∫0∞F(x)1/2-ɛ(1-F(x))1/2-ɛwF(F(x))dx
is finite. For this, by considering, for example, the two cases p≤1/2 and p>1/2 separately, we first easily verify the bound |wF(p)|≤c+clog(1/p)+clog(1/(1-p)). Hence, for every ɛ>0, there exists a constant c<∞ such that, for all p∈(0,1),
(7.5)|wF(p)|≤cpɛ(1-p)ɛ.
Bound (7.5) implies that integral (7.4) is finite when ∫0∞(1-F(x))1/2-2ɛdx<∞, which is true since the moment E[X2+α] is finite for some α>0 and the parameter ɛ>0 can be chosen as small as desired. Hence, n-1/2∑i=1nh(Xi)→dΓ with Γ denoting the integral on the right-hand side of statement (7.2). The random variable Γ is normal because the Brownian bridge ℬ(p) is a Gaussian process. Furthermore, Γ has mean zero because ℬ(p) has mean zero for every p∈[0,1]. The variance of Γ is equal to σF2 because E[ℬ(p)ℬ(q)]=min{p,q}-pq for all p,q∈[0,1]. We are left to show that E[Γ2]<∞. For this, we write the bound:
(7.6)E[Γ2]=∫0∞∫0∞E[ℬ(F(x))ℬ(F(y))]wF(F(x))wF(F(y))dxdy≤(∫0∞E[ℬ2(F(x))]wF(F(x))dx)2.
Since E[ℬ2(F(x))]=F(x)(1-F(x)), the finiteness of the integral on the right-hand side of bound (7.6) follows from the earlier proved statement that integral (7.4) is finite. Hence, E[Γ2]<∞ as claimed, which concludes the proof of Theorem 7.1.Theorem 7.2.
The empirical varianceSX,n2 is an estimator of σF2.Proof.
We construct an empirical estimator forσF2 by replacing every F on the right-hand side of (2.6) by the empirical Fn. Consequently, we replace the function wF(t) by its empirical version
(7.7)wX,n(t)=-∫0t(1p-1)ALn(p)(X̅-ALn(p))2dp+∫t1(1p-1)1X̅-ALn(p)dp.
We denote the resulting estimator of σF2 by SX,n2. The rest of the proof consists of verifying that this estimator coincides with the one defined by (2.8). Note that min{Fn(x),Fn(y)}-Fn(x)Fn(y)=0 when x∈[0,X1:n)∪[Xn:n,∞) and/or y∈[0,X1:n)∪[Xn:n,∞). Hence, the just defined SX,n2 is equal to
(7.8)∫X1:nXn:n∫X1:nXn:n(min{Fn(x),Fn(y)}-Fn(x)Fn(y))wX,n(Fn(x))wX,n(Fn(y))dxdy.
Since Fn(x)=k/n when x∈[Xk:n,Xk+1:n), we therefore have that
(7.9)SX,n2=∑k=1n-1∑l=1n-1(min{k,l}n-knln)×wX,n(kn)wX,n(ln)(Xk+1:n-Xk:n)(Xl+1:n-Xl:n).
Furthermore,
(7.10)wX,n(kn)=-∫0k/n(1p-1)ALn(p)(X̅-ALn(p))2dp+∫k/n1(1p-1)1X̅-ALn(p)dp=-∑i=1kIX,n(i)+∑i=k+1nJX,n(i),
where, using notations (6.6) and (6.8), the summands on the right-hand side of (7.10) are
(7.11)IX,n(i)=1Xi:n∫(i-1)/ni/n(1p-1)Λi,n+p(Ψi,n-p)2dp
for all i=1,…,n-1, and
(7.12)JX,n(i)=1Xi:n∫(i-1)/ni/n(1p-1)1Ψi,n-pdp
for all i=2,…,n. When i=1, then Λi,n=0. Hence, we immediately arrive at the expression for IX,n(1) given by (2.10). When 2≤i≤n-1, then
(7.13)IX,n(i)=Λi,nXi:nΨi,n2log(ii-1)-(Λi,n+Ψi,n)(Ψi,n-1)nXi:nΨi,n(Ψi,n-(i-1)/n)(Ψi,n-i/n)+1Xi:n(1+Λi,nΨi,n2)log(Ψi,n-(i-1)/nΨi,n-i/n),
and, after some algebra, we arrive at the right-hand side of (2.11). When 2≤i≤n-1, then we have the expression
(7.14)JX,n(i)=1Xi:nΨi,nlog(ii-1)-1Xi:n(1-1Ψi,n)log(Ψi,n-(i-1)/nΨi,n-i/n),
which, after some algebra, becomes the expression recorded in (2.12). When i=n, then Ψi,n=1, and so we see that JX,n(n) is given by (2.13). This completes the proof of Theorem 7.2.Theorem 7.3.
The empirical mixed momentSX,Y,n is an estimator of E[h(X)h(Y)].Proof.
We proceed similarly to the proof of Theorem7.2. We estimate the integrand P[X≤x,Y≤y]-F(x)H(y) using
(7.15)1n∑i=1n1{Xi≤x,Yi≤y}-1n∑i=1n1{Xi≤x}1n∑i=1n1{Yi≤y}.
After some rearrangement of terms, estimator (7.15) becomes
(7.16)1n∑i=1n1{Xi:n≤x,Y(i,n)≤y}-1n∑i=1n1{Xi:n≤x}1n∑i=1n1{Yi:n≤y}.
When x∈[Xk:n,Xk+1:n) and y∈[Yl:n,Yl+1:n), then estimator (7.16) is equal to n-1∑i=1k1{Y(i,n)≤Yl:n}-(k/n)(l/n), which leads us to the estimator SX,Y,n. This completes the proof of Theorem 7.3.


## 8. Proof of Theorem2.1

Throughout the proof we use the notationALF*(p) for the dual absolute Lorenz curve ∫p1F-1(t)dt, which is equal to μF-ALF(p). Likewise, we use the notation ALn*(p) for the empirical dual absolute Lorenz curve.Proof.
Simple algebra gives the equations(8.1)n(Z̃n-ZF)=-n∫01(1p-1)(ALn(p)ALn*(p)-ALF(p)ALF*(p))dp=-n∫01(1p-1)ALn(p)-ALF(p)ALF*(p)dp+n∫01(1p-1)ALF(p)ALF*2(p)(ALn*(p)-ALF*(p))dp+OP(rn,1)+OP(rn,2)
with the remainder terms
(8.2)rn,1=n∫01(1p-1)(ALn(p)-ALF(p))(1ALn*(p)-1ALF*(p))dp,rn,2=n∫01(1p-1)ALF(p)ALF*(p)(ALn*(p)-ALF*(p))(1ALn*(p)-1ALF*(p))dp.
We will later show (Lemmas 9.1 and 9.2) that the remainder terms rn,1 and rn,2 are of the order oP(1). Hence, we now proceed with our analysis of the first two terms on the right-hand side of (8.1), for which we use the (general) Vervaat process
(8.3)Vn(p)=∫0p(Fn-1(t)-F-1(t))dt+∫0F-1(p)(Fn(x)-F(x))dx
and its dual version
(8.4)Vn*(p)=∫p1(Fn-1(t)-F-1(t))dt+∫F-1(p)∞(Fn(x)-F(x))dx.
For mathematical and historical details on the Vervaat process, see Zitikis [23], Davydov and Zitikis [24], Greselin et al. [25], and references therein. Since ∫01(Fn-1(t)-F-1(t))dt=X̅-μF and ∫0∞(Fn(x)-F(x))dx=-(X̅-μF), adding the right-hand sides of (8.3) and (8.4) gives the equation Vn*(p)=-Vn(p). Hence, whatever upper bound we have for |Vn(p)|, the same bound holds for |Vn*(p)|. In fact, the absolute value can be dropped from |Vn(p)| since Vn(p) is always nonnegative. Furthermore, we know that Vn(p) does not exceed (p-Fn(F-1(p)))(Fn-1(p)-F-1(p)). Hence, with the notation en(p)=n(Fn(F-1(p))-p), which is the uniform on [0,1] empirical process, we have that
(8.5)nVn(p)≤|en(p)||Fn-1(p)-F-1(p)|.
Bound (8.5) implies the following asymptotic representation for the first term on the right-hand side of (8.1):
(8.6)-n∫01(1p-1)ALn(p)-ALF(p)ALF*(p)dp=n∫01(1p-1)1ALF*(p)(∫0F-1(p)(Fn(x)-F(x))dx)dp+OP(rn,3),
where
(8.7)rn,3=∫01(1p-1)1ALF*(p)|en(p)||Fn-1(p)-F-1(p)|dp.
We will later show (Lemma 9.3) that rn,3=oP(1). Furthermore, we have the following asymptotic representation for the second term on the right-hand side of (8.1):
(8.8)n∫01(1p-1)ALF(p)ALF*2(p)(ALn*(p)-ALF*(p))dp=-n∫01(1p-1)ALF(p)ALF*2(p)(∫F-1(p)∞(Fn(x)-F(x))dx)dp+OP(rn,4),
where
(8.9)rn,4=∫01(1p-1)ALF(p)ALF*2(p)|en(p)||Fn-1(p)-F-1(p)|dp.
We will later show (Lemma 9.4) that rn,4=oP(1). Hence, (8.1), (8.6) and (8.8) together with the aforementioned statements that rn,1,…,rn,4 are of the order oP(1) imply that
(8.10)n(Z̃n-ZF)=n∫01(1p-1)1ALF*(p)(∫0F-1(p)(Fn(x)-F(x))dx)dp-n∫01(1p-1)ALF(p)ALF*2(p)(∫F-1(p)∞(Fn(x)-F(x))dx)dp+oP(1)=1n∑i=1nh(Xi)+oP(1).
This completes the proof of Theorem 2.1.


## 9. Negligibility of Remainder Terms

The following four lemmas establish the above noted statements that the remainder termsrn,1,…,rn,4 are of the order oP(1). In the proofs of the lemmas we will use a parameter δ∈(0,1/2], possibly different from line to line but never depending on n. Furthermore, we will frequently use the fact that(9.1)E[Xq]<∞implies∫01|Fn-1(t)-F-1(t)|qdt=oP(1).
Another technical result that we will frequently use is the fact that, for any ɛ>0 as small as desired,(9.2)supx∈Rn|Fn(x)-F(x)|F(x)1/2-ɛ(1-F(x))1/2-ɛ=OP(1)
when n→∞.Lemma 9.1.
Under the conditions of Theorem2.1, rn,1=oP(1).Proof.
We split the remainder termrn,1=n∫01⋯dp into the sum of rn,1*(δ)=n∫01-δ⋯dp and rn,1**(δ)=n∫1-δ1⋯dp. The lemma follows if (1)
for everyδ>0, the statement rn,1*(δ)=oP(1) holds when n→∞,(2)
rn,1**(δ)=h(δ)OP(1) for a deterministic h(δ)↓0 when δ↓0, where OP(1) does not depend on δ.
To prove part (1), we first note that when0<p<1-δ, then ALF*(p)≥∫1-δ1F-1(t)dt, which is positive, and ALn*(p)≥∫1-δ1F-1(t)dt+oP(1) due to statement (9.1) with q=1. Hence, we are left to show that, when n→∞,
(9.3)n∫01-δ1p|ALn(p)-ALF(p)||ALn*(p)-ALF*(p)|dp=oP(1).
Since ALn*(p)-ALF*(p)=(X̅-μF)-(ALn(p)-ALF(p)), statement (9.3) follows if
(9.4)n|X̅-μF|∫01-δ1p|ALn(p)-ALF(p)|dp=oP(1),(9.5)n∫01-δ1p|ALn(p)-ALF(p)|2dp=oP(1).
We have n|X̅-μF|=OP(1) and |ALn(p)-ALF(p)|≤p(∫01|Fn-1(p)-F-1(p)|2dp)1/2. Since ∫01|Fn-1(p)-F-1(p)|2dp=oP(1) and ∫01-δp-1pdp<∞, we have statement (9.4). To prove statement (9.5), we use bound (8.5) and reduce the proof to showing that
(9.6)1n∫01-δ1p|∫0F-1(p)n(Fn(x)-F(x))dx|2dp=oP(1),(9.7)1n∫01-δ1p|en(p)|2|Fn-1(p)-F-1(p)|2dp=oP(1).
To prove statement (9.6), we use statement (9.2) and observe that
(9.8)∫01-δ1p(∫0F-1(p)F(x)1/2-ɛdx)2dp≤c(F,δ)∫01-δ1pp1-2ɛdp<∞.
To prove statement (9.7), we use the uniform on [0,1] version of statement (9.2) and Hölder's inequality, and in this way reduce the proof to showing that
(9.9)1n(∫01-δ1p2ɛadp)1/a(∫01-δ|Fn-1(p)-F-1(p)|2bdp)1/b=oP(1)
for some a,b>1 such that a-1+b-1=1. We choose the parameters a and b as follows. First, since E[X2+α]<∞, we set b=(2+α)/2. Next, we choose ɛ>0 on the left-hand side of statement (9.9) so that 2ɛa<1, which holds when ɛ<α/(4+2α) in view of the equation a-1+b-1=1. Hence, statement (9.9) holds and thus statement (9.7) follows. This completes the proof of part (1).
To establish part (2), we first estimate|rn,1**(δ)| from above using the bounds ALF*(p)≥(1-p)F-1(1/2) and ALn*(p)≥(1-p)Fn-1(1/2), which hold since δ≤1/2. Hence, we have reduced our task to verifying the statement n∫1-δ1|ALn(p)-ALF(p)|dp=h(δ)OP(1). Using the Vervaat process Vn(p) and bound (8.5), we reduce the proof of the statement to showing that the integrals
(9.10)∫1-δ1(∫0F-1(p)n|Fn(x)-F(x)|dx)dp,(9.11)∫1-δ1|en(p)||Fn-1(p)-F-1(p)|dp
are of the order h(δ)OP(1) with possibly different h(δ)↓0 in each case. In view of statement (9.2), we have the desired statement for integral (9.10) if the quantity
(9.12)∫1-δ1(∫0F-1(p)(1-F(x))1/2-ɛdx)dp
converges to 0 when δ↓0, in which case we use it as h(δ). The inner integral of (9.12) does not exceed ∫0∞(1-F(x))1/2-ɛdx, which is finite for all sufficiently small ɛ>0 since E[X2+α]<∞ for some α>0. This completes the proof that quantity (9.10) is of the order h(δ)OP(1). To show that quantity (9.11) is of a similar order, we use the uniform on [0,1] version of statement (9.2) and reduce the task to showing that ∫1-δ1|Fn-1(p)-F-1(p)|dp is of the order h(δ)OP(1). By the Cauchy-Bunyakowski-Schwarz inequality, we have that
(9.13)∫1-δ1|Fn-1(p)-F-1(p)|dp≤δ(∫01|Fn-1(p)-F-1(p)|2dp)1/2.
Since E[X2]<∞, we have ∫01|Fn-1(p)-F-1(p)|2dp=oP(1), and so setting h(δ)=δ establishes the desired asymptotic result for integral (9.11). This also completes the proof of part (2), and also of Lemma 9.1.Lemma 9.2.
Under the conditions of Theorem2.1, rn,2=oP(1).Proof.
Like in the proof of Lemma9.1, we split the remainder term rn,2=n∫01⋯dp into the sum of rn,2*(δ)=n∫01-δ⋯dp and rn,2**(δ)=n∫1-δ1⋯dp. To prove the lemma, we need to show the following. (1)
For everyδ>0, the statement rn,2*(δ)=oP(1) holds when n→∞.(2)
rn,2**(δ)=h(δ)OP(1) for a deterministic h(δ)↓0 when δ↓0, where OP(1) does not depend on δ.
To prove part (1), we first estimate|rn,2*(δ)| from above using the bounds p-1ALF(p)≤F-1(1-δ)<∞, ALF*(p)≥∫1-δ1F-1(t)dt>0, and ALn*(p)≥∫1-δ1F-1(t)dt+oP(1). This reduces our task to showing that, for every δ>0,
(9.14)n∫01-δ|ALn*(p)-ALF*(p)|2dp=oP(1).
Since ALn*(p)-ALF*(p)=(X̅-μF)-(ALn(p)-ALF(p)) and n(X̅-μF)2=oP(1), statement (9.14) follows from
(9.15)n∫01-δ|ALn(p)-ALF(p)|2dp=oP(1),
which is an elementary consequence of statement (9.5). This establishes part (1).
To prove part (2), we first estimate|rn,2**(δ)| from above using the bounds ALF*(p)≥(1-p)F-1(1/2) and ALn*(p)≥(1-p)Fn-1(1/2), and in this way reduce the task to showing that
(9.16)n∫1-δ111-p|ALn*(p)-ALF*(p)|dp=h(δ)OP(1).
Using the Vervaat process, statement (9.16) follows if
(9.17)∫1-δ111-p(∫F-1(p)∞n|Fn(x)-F(x)|dx)dp=h(δ)OP(1),(9.18)∫1-δ111-p|en(p)||Fn-1(p)-F-1(p)|dp=h(δ)OP(1)
with possibly different h(δ)↓0 in each case. Using statement (9.2), we have that statement (9.17) holds with h(δ) defined as the integral
(9.19)∫1-δ111-p(∫F-1(p)∞(1-F(x))1/2-ɛdx)dp,
which converges to 0 when δ↓0 as the following argument shows. First, we write the integrand as the product of (1-F(x))ɛ and (1-F(x))1/2-2ɛ. Then we estimate the first factor by (1-p)ɛ. The integral ∫0∞(1-F(x))1/2-2ɛdx is finite for all sufficiently small ɛ>0 since E[X2+α]<∞ for some α>0. Since ∫1-δ1(1-p)-1+ɛdp↓0 when δ↓0, integral (9.19) converges to 0 when δ↓0. The proof of statement (9.17) is finished.
We are left to prove statement (9.18). Using the uniform on [0,1] version of statement (9.2), we reduce the task to showing that
(9.20)∫1-δ11(1-p)1/2+ɛ|Fn-1(p)-F-1(p)|dp=h(δ)OP(1).
In fact, we will see below that OP(1) can be replaced by oP(1). Using Hölder's inequality, we have that the right-hand side of (9.20) does not exceed
(9.21)(∫1-δ11(1-p)(1/2+ɛ)adp)1/a(∫1-δ1|Fn-1(p)-F-1(p)|bdp)1/b
for some a,b>1 such that a-1+b-1=1. We choose the parameters a and b as follows. Since E[X2+α]<∞, we set b=2+α, and so the right-most integral of (9.21) is of the order oP(1). Furthermore, a=(2+α)/(1+α)<2, which can be made arbitrarily close to 2 by choosing sufficiently small α>0. Choosing ɛ>0 so small that (1/2+ɛ)a<1, we have that the left-most integral in (9.21) converges to 0 when δ↓0. This establishes statement (9.18) and completes the proof of Lemma 9.2.Lemma 9.3.
Under the conditions of Theorem2.1, rn,3=oP(1).Proof.
We split the remainder termrn,3=∫01⋯dp into the sum of rn,3*=∫01/2⋯dp and rn,3**=∫1/21⋯dp. The lemma follows if the two summands are of the order oP(1).
To provern,3*=oP(1), we use the bound ALF*(p)≥∫1/21F-1(p)dp and the uniform on [0,1] version of statement (9.2), and in this way reduce our task to showing that
(9.22)∫01/21p1/2+ɛ|Fn-1(p)-F-1(p)|dp=oP(1).
This statement can be established following the proof of statement (9.20), with minor modifications.
To provern,3**=oP(1), we use the bound ALF*(p)≥(1-p)F-1(1/2), the fact that supt|en(t)|=OP(1), and statement (9.1) with q=1. The desired result for rn,3** follows, which finishes the proof of Lemma 9.3.Lemma 9.4.
Under the conditions of Theorem2.1, rn,4=oP(1).Proof.
We splitrn,4=∫01⋯dp into the sum of rn,4*=∫01/2⋯dp and rn,4**=∫1/21⋯dp, and then show that the two summands are of the order oP(1).
To provern,4*=oP(1), we use the bounds p-1ALF(p)≤F-1(1/2)<∞ and ALF*(p)≥∫1/21F-1(p)dp>0 together with the uniform on [0,1] version of statement (9.2). This reduces our task to showing that ∫01/2|Fn-1(p)-F-1(p)|dp=oP(1), which holds due to statement (9.1) with q=1.
To provern,4**=oP(1), we use the bound ALF*(p)≥(1-p)F-1(1/2) and the uniform on [0,1] version of statement (9.2), and in this way reduce the proof to showing that
(9.23)∫1/211(1-p)1/2+ɛ|Fn-1(p)-F-1(p)|dp=oP(1).
This statement can be established following the proof of statement (9.20). The proof of Lemma 9.4 is finished.



---

*Source: 718905-2010-04-13.xml*