# Estimation of the Tail Index of Pareto-Type Distributions Using Regularisation

Authors: E. Ocran; R. Minkah; G. Kallah-Dagadu; K. Doku-Amponsah
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we introduce reduced-bias estimators for the estimation of the tail index of Pareto-type distributions. This is achieved through the use of a regularised weighted least squares with an exponential regression model for log-spacings of top-order statistics. The asymptotic properties of the proposed estimators are investigated analytically and found to be asymptotically unbiased, asymptotically consistent, and asymptotically normally distributed. Also, the finite sample behaviour of the estimators are studied through a simulation study The proposed estimators were found to yield low bias and mean square errors. In addition, the proposed estimators are illustrated through the estimation of the tail index of the underlying distribution of claims from the insurance industry.


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## Body

## 1. Introduction

Pareto-type distributions are often encountered in applications in the area of finance [1–3], reinsurance [4–6], risk management [7–9], and telecommunication [10, 11]. This distribution type has tail function(1)1−Fx=x−1/γℓFxasx⟶∞,or equivalently upper tail quantile function(2)Ux=xγℓuxasx⟶∞.The componentℓF and ℓu are slowly varying functions expressed as(3)limt⟶∞ℓxtℓt=1,for allx>0.The parameterγ is strictly positive for Pareto-type distributions and is also known as the tail index.SupposeX1,X2,…,Xn denote independent and identically distributed (i.i.d) random variables drawn from a distribution belonging to the maximum domain of attraction of the Pareto family of distributions, then for some auxiliary sequences of constants an>0;n≥1 and bn;n≥1 [12],(4)limn⟶∞PmaxX1,X2,…,Xn−bnan≤x=exp−1+γx−1/γ,1+γx≥0,where γ≥0. The estimation of γ continues to receive considerable attention in statistics of extremes as all inferences in extreme value analysis depend on the tail index. In practice, we seek estimators with less variance and bias as possible. A parametric or semiparametric approach can be employed to estimate the tail index, [13–16]. However, in this paper, we employ the semiparametric approach to develop reduced-bias estimators since they result in bias reduction.Under the semiparametric framework, the tail index estimators are dependent on thek largest observations, with these assumption about k:Assumption 1.
kn⟶∞ as n⟶∞.Assumption 2.
k=kn=On as n⟶∞.
The most widely used semiparametric tail index estimator is the Hill estimator [17]. The author in [17] approximates the top k order statistics with a Pareto distribution and estimates γ>0 using a maximum likelihood estimator (MLE). The Hill estimator has the minimum asymptotic variance among the semiparametric estimators but it is very sensitive to the choice of k [18]. This drawback of the estimator makes its usage challenging in practice, especially in the selection of the tail fraction, k. The author in [17] defined the tail estimator as(5)γ^kH=1k∑j=1klogXn−j+1,nXn−k,n.
The Hill estimator due to its popularity has received several generalisations: (see, for example, the works of [19–27]). Even though, these estimators possesses desirable properties of a good estimator, they are sensitive to changes in k. In practice, this possesses a problem in choosing which value of k is to be chosen for the estimation of tail index and other extreme value parameters. In view of this, several authors have looked at the selection of optimal values of k (see [28] for a comprehensive review on threshold selection).
The authors in [29] proposed the bias-corrected Hill estimators (which a refinement of the Hill estimator) to solve the challenges of the Hill estimator to some extent. Specifically, the authors proposed two approaches for reducing the bias of the Hill estimator while maintaining the asymptotic variance of the Hill estimator. Empirically, the bias-corrected Hill estimator yields stable tail index estimates compared to the Hill estimator, i.e., the bias-corrected Hill estimator is less sensitive to the choice of k relative to the Hill estimator. However, this estimator appears to be unstable for large values of k.
In this study, we seek to propose alternative tail index estimators that yield much more stable tail index estimates relative to the number of top-order statistics and attain the minimum asymptotic variance of the Hill estimator under some conditions. The former is of much importance in the extreme value literature in addressing the problem of selection of tail sample fraction for semiparametric estimators of tail index and extreme events. The proposed method employs the regularised weighted least squares method, which entails weighting the data to account for the variability in the data and penalising the bias term. This technique minimises the bias reduction effect for smallerk, resulting in bias-reduced estimators that attain the Hill estimator’s asymptotic variance.


## 2. Estimation Methods

We letX1,X2,X3,…,Xn denote a sequence of i.i.d random variables drawn from a population with distribution function F and the associated tail quantile function U. Let X1,n≤X2,n≤⋯≤Xn,n be the order statistics associated with the sample. Using equation (2), the order statistics can be jointly expressed as(6)logXn−j+1,n~dγlogUj,n−1+logℓUj,n−1,where Uj,n−1, j=1,2,3,…,n represent the order statistics of the standard uniform distribution. Using equation (6), the authors in [30] demonstrated that(7)logXn−j+1,nlogXn−k,n~dγlogUk+1,n−1Uj,n−1+logℓUj,n−1Uk+1,n−1,where k∈2,3,…,n−1 and also obtained a more refined expression of equation (7) by imposing a second-order assumption on the rate of convergence to equation (3). This is stated as an assumption as follows:Assumption 3.
There exists a real constantρ<0 and a rate function b that is regularly varying, with index ρ, i.e., bx=xρℓbx for some ℓb satisfying equation (3) for all v≥1:(8)limx⟶∞logℓuxvℓux=bxhρv,with hρv=∫1vyρ−1dy [30].
Under Assumption3, the authors in [30] showed that the weighted log-spacings of order statistics(9)Tj=jlogXn−j+1,n−logXn−j,n,1≤j≤k<n,are approximately exponentially distributed. They particularly obtained the expression(10)Tj=γ+bn,kjk+1−ρfj,1≤j≤k,where bn,k=bn+1/k+1⟶0 as k,n⟶∞, and fj′s are i.i.d exponentially distributed with a unit mean and ρρ<0 is a second-order parameter. The authors employed MLE to the estimate the parameters in equation (10).
Using equation (10) and Assumptions 1 and 2, the authors in [31] demonstrated that Tj can be further approximated as a regression model(11)Tj=Tjρ,εj=γ+bn,kCjρ+εjfor,j∈1,2,…,k,where bn,k=bn/k is the slope, Cj=Cjρ=j/k+1−ρ is the covariate, γ is the intercept, and εj is the error terms with asymptotic mean, 0, and variance, γ2.
The authors in [31] proposed the ordinary least squares estimator for the estimation of γ in equation (11). Furthermore, based on equation (11), the authors in [13] have introduced the ridge regression estimator for estimating γ. In this paper, we propose the regularised weighted least squares estimators for estimating γ in equation (11).


### 2.1. The Proposed Estimators

In order to estimateγ, the loss function of the regularised weighted least squares for equation (11) is defined as(12)Lkγ,bn,k;λ,W=∑j=1kWjTj−γ−bn,kCj2+λkbn,k2,λ≥0.Here,Wj is the weight function defined as(13)Wj≔Wjθj=1−θjαkjk+1,j∈1,2,…,k,whereθj∼i.i.dU0,1. Thus, Wj∈0,1 and decreases linearly with respect to j. Wj is employed due to its ability to reduce the variability in the estimator. The exponent αk≥0 is chosen such that(14)Δ≔limk⟶∞αkk<∞.In this study, we consider 0<Δ≤1. Thus, we would define αk such that 0<αk≤k. Note that, Wj is random through θj, and when the exponent is 0, Wj is deterministic. In particular, when αk=0, we obtain the weight function gj=1−j/k+1,j∈1,2,…,k, as introduced in [32]. Nevertheless, we can approximate the weight gj as a limit of the current result by allowing Δ to approach 0.We minimize the loss functionLk in equation (12) with respect to γ and bn,k to obtain jointly estimates γ^ and b^n,k(15)b^n,kλ,θ=∑j=1kWj˜Cj−∑j=1kW˜jCjTj2καkλ+∑j=1kWj˜Cj2−∑j=1kWj˜Cj2,and(16)γ^RWλλ,θ=∑j=1kWj˜Tj−b^n,kλ,θ∑j=1kWj˜Cj.while,(17)Wj˜=Wj∑j=1kWj,1≤j≤k,(18)Cj=jk+1−ρ,forj∈1,2,…,k,ρ<0,and(19)καk=αk+12αk+1.We substitute equation (13) into equation (17) to obtain an explicit expression for equation (17) as(20)Wj˜≈2καkk1−θjαkjk+1.The main theorem backing the proof of equation (20) is the Kolmogorov’s strong law of large numbers for independent random variables, but for brevity, we present only the results.The parameterρ<0 is estimated externally using the minimum variance approach introduced in [13].In addition, the parameterλ in equation (12) is the penalty that regulates the bias coefficient bn,k. The loss function, Lk, minimises the weighted sum of squared residuals and also regulates the size of the bias coefficient bn,k. The penalty term shrinks the bias term, bn,k to 0 as the penalty parameter, λ, increases. Thus, the larger the value of λ, the higher the contribution of the penalty term to the loss function and the stronger the regularisation process. To obtain an estimator for the penalty term, λ, we minimize the asymptotic mean squared error (AMSE) of the proposed estimator, γ^RWλ (see, example [13]).Note that, since the weight function depends on theθj’s, the estimators in equations (15) and (16) also depend on the θj’s through the weight function. Therefore, we would find the AMSE by conditioning on the θj’s. From equation (15) and (16), the AMSE for γ^RWλ is obtained as(21)AMSEγ^RWλλ|θj=θ=γ243kϕαk+2S1θS˙θ2καkλ+S2θ+S12θS¨θ2καkλ+S2θ2+2bn,kλκαkS1θλ2καkλ+S2θ2,where(22)S1θ=∑j=1kW˜jCj,(23)S2θ=∑j=1kW˜jCj2−∑j=1kWj˜Cj2,(24)S˙θ=∑j=1kW˜j2S1θ−Cj,(25)S¨θ=∑j=1kW˜j2S1θ−Cj2,and(26)ϕαk=αk+16α2k+4αk+12αk+12.We now derive an expression for the penalty term,λ. Minimising AMSEγ^RWλ|θj=θ in equation (21) over λ, the optimal value of λ is obtained by solving the equation(27)2bn,k2καkS1θS2θλ−2καkγ2S˙θλ−γ2S2θS˙θ−γ2S1θS¨θ=0.We obtain,(28)λkθ=γ2S2θS˙θ+γ2S1θS¨θ2bn,k2καkS1θS2θ−2καkγ2S˙θ.In order to estimateλ^kθ, we assume the slowly varying function ℓb in (8) is constant [13]. Thus, we have(29)bx=γβxρ1+o1ask⟶∞,for some β∈ℝ and we estimate β via the estimator proposed in [33]. It follows from equation (28) and (29) that(30)λ^kθ=S1θS¨θ+S˙θS2θ2καkS1θS2θβ^k2n/k2ρ^−2καkS˙θ.The penalty term,λ is required to be non-negative; therefore, we define λ^k+θ=maxλ^kθ,0. We then obtain a penalty term and estimators which do not depend on θj’s, by averaging λ^kθ over the θj’s, as follows:(31)λ^k=Eθλ^kθ1S1θS2θβ^kn/k2ρ^>S˙θ,and we defined the proposed estimator ofγ by(32)γRWλ≔EθγRWλ,θ.


### 2.2. Asymptotic Properties of the Proposed Estimators

Unbiasedness, consistency, and normality are desirable properties of a good estimator. In this section, we investigate these desirable properties of the proposed estimators.We shall summarise the asymptotic behaviour of the statistics used to build the AMSE of the proposed estimator in Lemma1. These properties will be required in the proof of the asymptotic consistency and sampling distribution of the proposed estimator. Henceforth, anytime we use the term a.s it is with respect to the law of the i.i.d sequence θ1,θ2,θ3,…,θk.Lemma 1.
Assume thatρ<0 is estimated by a consistent estimator ρ^ and (14) holds, then as k⟶∞ and k/n⟶0;(i)
S1θ=∑j=1kW˜jCj⟶a.s1/1−ρ.(ii)
S2θ=∑j=1kW˜jCj2−∑j=1kW˜jCj2⟶a.sρ2/1−2ρ1−ρ2.(iii)
S˙θ=∑j=1kW˜j2S1θ−Cj⟶a.s0.(iv)
S¨θ=∑j=1kW˜j2S1θ−Cj2⟶a.s0.Lemma 2.
Supposeρ<0 and β∈ℝ are estimated by their respective consistent estimators ρ^ and βk^, then as k,n⟶∞ and k/n⟶0,(33)λ^kθ⟶a.s0.It follows from Lemma2 that the regularised weighted least estimator, γ^RWλ is asymptotically unbiased. That is, as k⟶∞, biasγ^RWλ⟶0. The bias of the proposed estimator is given by(34)biasγ^RWλ=EEθγRWθ,λ−γ=Eθ2bn,kκαkS1θλθ2καkλθ+S2θ,where(35)biasγ^RWλ|θ=θ^=2bn,kκαkS1θ^λθ2καkλθ^+S2θ^.Since the term in the bracket converges to 0 ask⟶∞ almost sure by Lemma 2, the expectation of the term will converge to 0 as k⟶∞. Therefore,(36)limk⟶∞biasγ^RWλ=limk⟶∞∫0,1k2bn,kκαkS1θλθ2καkλθ+S2θdθ=∫0,1∞limk⟶∞2bn,kκαkS1θλθ2καkλθ+S2θdθ=0,which gives limk⟶∞biasγ^RWλ=0. Similarly, we can use Lemma 1 and Lemma 2 to show that AMSEγ^RWλ⟶0 as k⟶∞. We write(37)AMSEγ^RWθ,λ|θ≔γ243kϕαk+2S1θS˙θ2καkλ+S2θ+S12θS¨θ2καkλ+S2θ2+biasγ^RWλ,θ|θ2.Now, observe from the Jensen’s inequality (applied to the expectation taken with respect to the law ofθ) and the Fubini’s theorem that(38)Eθbiasγ^RWθ,λ|θ2≤AMSEγ^RWλ≤EθEγ^RWθ,λ−Eγ^RWθ,λ2+Eθbiasγ^RWλ,θ|θ2=EθAMSEγ^RWλ,θ|θ,and therefore, we have(39)0≤limk⟶∞AMSEγ^RWλ≤limk⟶∞∫0,1kAMSEγ^RWλ|θ^=θdθ=∫0,1∞limk⟶∞AMSEγ^RWλ|θ^=θdθ=0.This implies that the proposed estimator is asymptotically consistent under some conditions.Theorem 3.
Suppose equations (2), (8), and (14) are satisfied. Assume also that ρ is estimated by a consistent estimator ρ^ with(40)E1limk⟶∞S2ρ^=E1−2ρ^1−ρ^2ρ^2<∞.Then, if assumptions A1 and A2 holds, and kbn,k⟶p0, then, we have(41)kEθγ^RWλ,θ−γ⟶dN0,γ2.
Theorem3 discusses the asymptotic normality of γ^RWλ defined in equation (16). To prove Theorem 3, we require the following properties in addition.We write(42)Mλθ≔W˜jCj−∑j=1kW˜jCj2καkλ+∑j=1kW˜jCj2−∑j=1kW˜jCj2.Lemma 4.
LetCj=j/k+1−ρ,W˜j=Wj/∑j=1kWj,j∈1,2,…,k and ρ<0. Then, as k⟶∞,(43)Mλθ⟶O1k1+ω,0<ω≤0.1.
Lemma4 is required in the Proof Proof 5.Lemma 5.
LetT1,T2,T3,… be independent random variables from an exponential distribution with mean μi<∞, for all i. Then, for any ϵ>0,(44)limk⟶∞ℙkb^n,k≥ε=0,where b^n,k is defined by equation (15).Remark 1.
Lemma5 shows the statistics kb^n,k⟶p0 as k,n⟶∞ and k/n⟶0.
The next lemma is about the satisfaction of the Lyapunov’s version of the central limit theorem. The Lyapunov’s variant of the central limit theorem assumes the existence of a finite moment of an order higher than two.Lemma 7.
Suppose thatZ1,Z2,… are independent random variables such that EZk=μ^k and VarZk=σk2<∞, then, there exists δ>0 such that(45)limk⟶∞1sn2+δ∑j=1kEYj−μ^j2+δ=0,where Zk=EθW˜kθkTk.Remark 2.
Setting the penalty term to 0 reduces the regularised weighted least squares estimator to a weighted least squares estimator. The difference between this weighted least squares estimator and the one introduced by [32] is that, this weighted least squares estimator has smaller asymptotic variance and this is due to the introduction of randomness into the weight function. The resulting weighted least squares estimator is also asymptotically unbiased, asymptotically consistent, and asymptotically normally distributed with mean 0 and variance γ2.


## 2.1. The Proposed Estimators

In order to estimateγ, the loss function of the regularised weighted least squares for equation (11) is defined as(12)Lkγ,bn,k;λ,W=∑j=1kWjTj−γ−bn,kCj2+λkbn,k2,λ≥0.Here,Wj is the weight function defined as(13)Wj≔Wjθj=1−θjαkjk+1,j∈1,2,…,k,whereθj∼i.i.dU0,1. Thus, Wj∈0,1 and decreases linearly with respect to j. Wj is employed due to its ability to reduce the variability in the estimator. The exponent αk≥0 is chosen such that(14)Δ≔limk⟶∞αkk<∞.In this study, we consider 0<Δ≤1. Thus, we would define αk such that 0<αk≤k. Note that, Wj is random through θj, and when the exponent is 0, Wj is deterministic. In particular, when αk=0, we obtain the weight function gj=1−j/k+1,j∈1,2,…,k, as introduced in [32]. Nevertheless, we can approximate the weight gj as a limit of the current result by allowing Δ to approach 0.We minimize the loss functionLk in equation (12) with respect to γ and bn,k to obtain jointly estimates γ^ and b^n,k(15)b^n,kλ,θ=∑j=1kWj˜Cj−∑j=1kW˜jCjTj2καkλ+∑j=1kWj˜Cj2−∑j=1kWj˜Cj2,and(16)γ^RWλλ,θ=∑j=1kWj˜Tj−b^n,kλ,θ∑j=1kWj˜Cj.while,(17)Wj˜=Wj∑j=1kWj,1≤j≤k,(18)Cj=jk+1−ρ,forj∈1,2,…,k,ρ<0,and(19)καk=αk+12αk+1.We substitute equation (13) into equation (17) to obtain an explicit expression for equation (17) as(20)Wj˜≈2καkk1−θjαkjk+1.The main theorem backing the proof of equation (20) is the Kolmogorov’s strong law of large numbers for independent random variables, but for brevity, we present only the results.The parameterρ<0 is estimated externally using the minimum variance approach introduced in [13].In addition, the parameterλ in equation (12) is the penalty that regulates the bias coefficient bn,k. The loss function, Lk, minimises the weighted sum of squared residuals and also regulates the size of the bias coefficient bn,k. The penalty term shrinks the bias term, bn,k to 0 as the penalty parameter, λ, increases. Thus, the larger the value of λ, the higher the contribution of the penalty term to the loss function and the stronger the regularisation process. To obtain an estimator for the penalty term, λ, we minimize the asymptotic mean squared error (AMSE) of the proposed estimator, γ^RWλ (see, example [13]).Note that, since the weight function depends on theθj’s, the estimators in equations (15) and (16) also depend on the θj’s through the weight function. Therefore, we would find the AMSE by conditioning on the θj’s. From equation (15) and (16), the AMSE for γ^RWλ is obtained as(21)AMSEγ^RWλλ|θj=θ=γ243kϕαk+2S1θS˙θ2καkλ+S2θ+S12θS¨θ2καkλ+S2θ2+2bn,kλκαkS1θλ2καkλ+S2θ2,where(22)S1θ=∑j=1kW˜jCj,(23)S2θ=∑j=1kW˜jCj2−∑j=1kWj˜Cj2,(24)S˙θ=∑j=1kW˜j2S1θ−Cj,(25)S¨θ=∑j=1kW˜j2S1θ−Cj2,and(26)ϕαk=αk+16α2k+4αk+12αk+12.We now derive an expression for the penalty term,λ. Minimising AMSEγ^RWλ|θj=θ in equation (21) over λ, the optimal value of λ is obtained by solving the equation(27)2bn,k2καkS1θS2θλ−2καkγ2S˙θλ−γ2S2θS˙θ−γ2S1θS¨θ=0.We obtain,(28)λkθ=γ2S2θS˙θ+γ2S1θS¨θ2bn,k2καkS1θS2θ−2καkγ2S˙θ.In order to estimateλ^kθ, we assume the slowly varying function ℓb in (8) is constant [13]. Thus, we have(29)bx=γβxρ1+o1ask⟶∞,for some β∈ℝ and we estimate β via the estimator proposed in [33]. It follows from equation (28) and (29) that(30)λ^kθ=S1θS¨θ+S˙θS2θ2καkS1θS2θβ^k2n/k2ρ^−2καkS˙θ.The penalty term,λ is required to be non-negative; therefore, we define λ^k+θ=maxλ^kθ,0. We then obtain a penalty term and estimators which do not depend on θj’s, by averaging λ^kθ over the θj’s, as follows:(31)λ^k=Eθλ^kθ1S1θS2θβ^kn/k2ρ^>S˙θ,and we defined the proposed estimator ofγ by(32)γRWλ≔EθγRWλ,θ.


## 2.2. Asymptotic Properties of the Proposed Estimators

Unbiasedness, consistency, and normality are desirable properties of a good estimator. In this section, we investigate these desirable properties of the proposed estimators.We shall summarise the asymptotic behaviour of the statistics used to build the AMSE of the proposed estimator in Lemma1. These properties will be required in the proof of the asymptotic consistency and sampling distribution of the proposed estimator. Henceforth, anytime we use the term a.s it is with respect to the law of the i.i.d sequence θ1,θ2,θ3,…,θk.Lemma 1.
Assume thatρ<0 is estimated by a consistent estimator ρ^ and (14) holds, then as k⟶∞ and k/n⟶0;(i)
S1θ=∑j=1kW˜jCj⟶a.s1/1−ρ.(ii)
S2θ=∑j=1kW˜jCj2−∑j=1kW˜jCj2⟶a.sρ2/1−2ρ1−ρ2.(iii)
S˙θ=∑j=1kW˜j2S1θ−Cj⟶a.s0.(iv)
S¨θ=∑j=1kW˜j2S1θ−Cj2⟶a.s0.Lemma 2.
Supposeρ<0 and β∈ℝ are estimated by their respective consistent estimators ρ^ and βk^, then as k,n⟶∞ and k/n⟶0,(33)λ^kθ⟶a.s0.It follows from Lemma2 that the regularised weighted least estimator, γ^RWλ is asymptotically unbiased. That is, as k⟶∞, biasγ^RWλ⟶0. The bias of the proposed estimator is given by(34)biasγ^RWλ=EEθγRWθ,λ−γ=Eθ2bn,kκαkS1θλθ2καkλθ+S2θ,where(35)biasγ^RWλ|θ=θ^=2bn,kκαkS1θ^λθ2καkλθ^+S2θ^.Since the term in the bracket converges to 0 ask⟶∞ almost sure by Lemma 2, the expectation of the term will converge to 0 as k⟶∞. Therefore,(36)limk⟶∞biasγ^RWλ=limk⟶∞∫0,1k2bn,kκαkS1θλθ2καkλθ+S2θdθ=∫0,1∞limk⟶∞2bn,kκαkS1θλθ2καkλθ+S2θdθ=0,which gives limk⟶∞biasγ^RWλ=0. Similarly, we can use Lemma 1 and Lemma 2 to show that AMSEγ^RWλ⟶0 as k⟶∞. We write(37)AMSEγ^RWθ,λ|θ≔γ243kϕαk+2S1θS˙θ2καkλ+S2θ+S12θS¨θ2καkλ+S2θ2+biasγ^RWλ,θ|θ2.Now, observe from the Jensen’s inequality (applied to the expectation taken with respect to the law ofθ) and the Fubini’s theorem that(38)Eθbiasγ^RWθ,λ|θ2≤AMSEγ^RWλ≤EθEγ^RWθ,λ−Eγ^RWθ,λ2+Eθbiasγ^RWλ,θ|θ2=EθAMSEγ^RWλ,θ|θ,and therefore, we have(39)0≤limk⟶∞AMSEγ^RWλ≤limk⟶∞∫0,1kAMSEγ^RWλ|θ^=θdθ=∫0,1∞limk⟶∞AMSEγ^RWλ|θ^=θdθ=0.This implies that the proposed estimator is asymptotically consistent under some conditions.Theorem 3.
Suppose equations (2), (8), and (14) are satisfied. Assume also that ρ is estimated by a consistent estimator ρ^ with(40)E1limk⟶∞S2ρ^=E1−2ρ^1−ρ^2ρ^2<∞.Then, if assumptions A1 and A2 holds, and kbn,k⟶p0, then, we have(41)kEθγ^RWλ,θ−γ⟶dN0,γ2.
Theorem3 discusses the asymptotic normality of γ^RWλ defined in equation (16). To prove Theorem 3, we require the following properties in addition.We write(42)Mλθ≔W˜jCj−∑j=1kW˜jCj2καkλ+∑j=1kW˜jCj2−∑j=1kW˜jCj2.Lemma 4.
LetCj=j/k+1−ρ,W˜j=Wj/∑j=1kWj,j∈1,2,…,k and ρ<0. Then, as k⟶∞,(43)Mλθ⟶O1k1+ω,0<ω≤0.1.
Lemma4 is required in the Proof Proof 5.Lemma 5.
LetT1,T2,T3,… be independent random variables from an exponential distribution with mean μi<∞, for all i. Then, for any ϵ>0,(44)limk⟶∞ℙkb^n,k≥ε=0,where b^n,k is defined by equation (15).Remark 1.
Lemma5 shows the statistics kb^n,k⟶p0 as k,n⟶∞ and k/n⟶0.
The next lemma is about the satisfaction of the Lyapunov’s version of the central limit theorem. The Lyapunov’s variant of the central limit theorem assumes the existence of a finite moment of an order higher than two.Lemma 7.
Suppose thatZ1,Z2,… are independent random variables such that EZk=μ^k and VarZk=σk2<∞, then, there exists δ>0 such that(45)limk⟶∞1sn2+δ∑j=1kEYj−μ^j2+δ=0,where Zk=EθW˜kθkTk.Remark 2.
Setting the penalty term to 0 reduces the regularised weighted least squares estimator to a weighted least squares estimator. The difference between this weighted least squares estimator and the one introduced by [32] is that, this weighted least squares estimator has smaller asymptotic variance and this is due to the introduction of randomness into the weight function. The resulting weighted least squares estimator is also asymptotically unbiased, asymptotically consistent, and asymptotically normally distributed with mean 0 and variance γ2.


## 3. Simulation Study

In the previous section, we proposed the regularised weighted least squares estimators under the semiparametric setting to estimate the tail index of the underlying distribution of a given data from the Pareto-type of distributions. In this section, we perform a simulation study to compare the performance of our proposed estimators to other existing semiparametric tail index estimators. Particularly, the regularised weighted least squares, RWLS, the reduced-bias weighted least squares with modified weight function, WLS, the ridge regression, RR [13], the least squares, LS [31], the Hill estimator, HILL [17], and the bias-corrected Hill, BCHILL [29] in the case of Pareto-type distributions are compared.


### 3.1. Simulation Design

We consider the Fréchet and Burr XII from the Pareto-type distributions as shown in Table1. For each distribution F, we generate 1000 repetitions of samples of size n=50,500, and 2000. For the Fréchet distributions, we consider α=10,2, and 1.0; and for the Burr XII we consider the mixtures.(i)
ξ=10&τ=10,(ii)
ξ=2&τ=2 and(iii)
ξ=2&τ=1/2Table 1
Heavy-tailed distributions from the Pareto-type distribution.
Distribution1−FxℓFxγBurr type XII1+xτ−ξ1+x−τ−ξ1/τξFréchet1−Fx=1−exp−x−α1−x−α/2+Ox−α1/αTo obtain the tail index valuesγ=0.1,0.5 and 1.0, respectively. We consider the finite sample behaviour of the proposed estimators, RWLS and WLS, and also compared these estimators with RR, LS, BCHILL and HILL. The mean square errors (MSE) and bias are plotted as a function of the number of top-order statistics, k, to investigate the estimators’ sample path behaviour.In the case of the weight function, theθj’s will be replaced with their point estimate, in this case, the mean of a standard uniform distribution. In the case of αk, we select αk such that Δ⟶0, as k⟶∞. This choice of αk is made because in practice we have observed that it yields much more stable estimates compared to when αk is selected such that Δ⟶c>0 in application.


### 3.2. Discussion of Simulation Results

In this section, we discuss the behaviour of RWLS and WLS relative to RR, LS, HILL, and BCHILL. The MSE and bias are the performance measures in the simulation studies. The simulation results for the Burr distribution with different tail indexes are shown in Figures1–3. Also, Figures 4–6 present the simulation results for the Fréchet distribution with varying tail indexes.Figure 1
Results for Burr type XII distribution withγ=0.1: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 2
Results for Burr type XII distribution withγ=0.5: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 3
Results for Burr type XII distribution withγ=1.0: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 4
Results for Fréchet distribution withγ=0.1: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 5
Results for Fréchet distribution withγ=0.5: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 6
Results for Fréchet distribution withγ=1.0: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.From these figures, the plots of WLS and RWLS follow the same sample path fork≤0.4n, i.e., their performance are relatively the same on that interval. WLS and LS are very close to each other, though generally, WLS slightly outperforms LS in terms of MSE and bias. Thus, generally the WLS can be considered the most appropriate estimator of the tail index among the regression-based estimators (i.e., RR, LS, WLS, and RWLS) since it mostly has smallest bias and MSE across all samples.Additionally, the MSE plots of the proposed estimators are low and near constant over the central part ofk, except in the case of Burr XII with γ=1.0. With the exception of the HILL estimator (which globally has the highest MSE), the MSE curves of the estimators are mostly close to each other in the central k region, especially in the case of the Fréchet distribution. This implies that the proposed estimators are competitive with the existing estimators. However, the proposed estimators, WLS and RWLS, generally attain the lowest bias for small samples, i.e., n=50. Furthermore, for medium to large values of k, the sample paths of RWLS in the MSE and bias plots are between HILL and RR. Even though the BCHILL estimator mostly has the smallest MSE and bias, the proposed estimators (RWLS and WLS) outperform it for large values of k.Hence, from the simulation results, WLS and RWLS are appropriate alternatives for the estimation of the tail index of the Pareto-type distributions in terms of MSE and bias.In addition, no single tail index estimator under investigation was found to be universally the best in terms of the MSE and bias. Finally, theR codes for the simulation and the application studies are available at https://github.com/kikiocran/RegularisedTail.


## 3.1. Simulation Design

We consider the Fréchet and Burr XII from the Pareto-type distributions as shown in Table1. For each distribution F, we generate 1000 repetitions of samples of size n=50,500, and 2000. For the Fréchet distributions, we consider α=10,2, and 1.0; and for the Burr XII we consider the mixtures.(i)
ξ=10&τ=10,(ii)
ξ=2&τ=2 and(iii)
ξ=2&τ=1/2Table 1
Heavy-tailed distributions from the Pareto-type distribution.
Distribution1−FxℓFxγBurr type XII1+xτ−ξ1+x−τ−ξ1/τξFréchet1−Fx=1−exp−x−α1−x−α/2+Ox−α1/αTo obtain the tail index valuesγ=0.1,0.5 and 1.0, respectively. We consider the finite sample behaviour of the proposed estimators, RWLS and WLS, and also compared these estimators with RR, LS, BCHILL and HILL. The mean square errors (MSE) and bias are plotted as a function of the number of top-order statistics, k, to investigate the estimators’ sample path behaviour.In the case of the weight function, theθj’s will be replaced with their point estimate, in this case, the mean of a standard uniform distribution. In the case of αk, we select αk such that Δ⟶0, as k⟶∞. This choice of αk is made because in practice we have observed that it yields much more stable estimates compared to when αk is selected such that Δ⟶c>0 in application.


## 3.2. Discussion of Simulation Results

In this section, we discuss the behaviour of RWLS and WLS relative to RR, LS, HILL, and BCHILL. The MSE and bias are the performance measures in the simulation studies. The simulation results for the Burr distribution with different tail indexes are shown in Figures1–3. Also, Figures 4–6 present the simulation results for the Fréchet distribution with varying tail indexes.Figure 1
Results for Burr type XII distribution withγ=0.1: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 2
Results for Burr type XII distribution withγ=0.5: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 3
Results for Burr type XII distribution withγ=1.0: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 4
Results for Fréchet distribution withγ=0.1: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 5
Results for Fréchet distribution withγ=0.5: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.Figure 6
Results for Fréchet distribution withγ=1.0: Bias (top row) and MSE (bottom row). First column: n=50; second column: n=500; and third column: n=2000.From these figures, the plots of WLS and RWLS follow the same sample path fork≤0.4n, i.e., their performance are relatively the same on that interval. WLS and LS are very close to each other, though generally, WLS slightly outperforms LS in terms of MSE and bias. Thus, generally the WLS can be considered the most appropriate estimator of the tail index among the regression-based estimators (i.e., RR, LS, WLS, and RWLS) since it mostly has smallest bias and MSE across all samples.Additionally, the MSE plots of the proposed estimators are low and near constant over the central part ofk, except in the case of Burr XII with γ=1.0. With the exception of the HILL estimator (which globally has the highest MSE), the MSE curves of the estimators are mostly close to each other in the central k region, especially in the case of the Fréchet distribution. This implies that the proposed estimators are competitive with the existing estimators. However, the proposed estimators, WLS and RWLS, generally attain the lowest bias for small samples, i.e., n=50. Furthermore, for medium to large values of k, the sample paths of RWLS in the MSE and bias plots are between HILL and RR. Even though the BCHILL estimator mostly has the smallest MSE and bias, the proposed estimators (RWLS and WLS) outperform it for large values of k.Hence, from the simulation results, WLS and RWLS are appropriate alternatives for the estimation of the tail index of the Pareto-type distributions in terms of MSE and bias.In addition, no single tail index estimator under investigation was found to be universally the best in terms of the MSE and bias. Finally, theR codes for the simulation and the application studies are available at https://github.com/kikiocran/RegularisedTail.


## 4. Applications

In this section, we consider the estimation of the tail index of the underlying distribution of two datasets from the insurance industry. First, the SOA Group Medical Insurance dataset which consists of over 170,000 claims recorded from 1991 to 1992. In this study, we consider the 1991 dataset, which comprised 75,789 claims and have been studied widely in the extreme value context (see, for example, [4, 18]). Considering the large size of this dataset, we focus on the extreme tail of the data and hence consider the top 10% data points, (i.e, 0<k≤0.1n). The SOA dataset is available at https://lstat.kuleuven.be/Wiley/Data/soa.txt.Second, an automobile insurance data from Ghana which consists of 452 claims from July 7, 2020, to May 11, 2021, and can be found athttps://github.com/kikiocran/RegularisedTail. We will refer to this dataset as the GH claims in this study. To the best of our knowledge, this dataset has never been used in the extreme value theory literature.The scatter plots of the SOA, and the GH claims are shown in Figure7. We observe that two claims and one claim in the SOA and GH claims, respectively, appear to be far detached from the bulk of the data. These observations can also be seen to deviate from linearity and far removed from the bulk of the points, respectively, in the Pareto and exponential Q-Q plots (Figure 8) of the two datasets. Such large observations are suspected outliers and may significantly influence the tail index estimates (see, for example, [4]). The convex curvature of the exponential Q-Q plots and the near linearity of the Pareto Q-Q plots of the datasets indicate the datasets suggest they belong to the Pareto-type distributions.Figure 7
Scatter plots: SOA claims data (left panel); GH claims data (right panel).Figure 8
Exponential and Pareto Q-Q plots: SOA claims (left panel); GH claims (right panel).Figure9 shows the sample paths of the tail index estimators for the underlying distributions of the two datasets. The plot of HILL diverges as k increases, i.e., it is very sensitive to the changes in k. Hence, it is not an appropriate estimator for estimating the tail index. The other estimators exhibit some form of stability; however, the sample paths of the proposed estimators (i.e., RWLS and WLS) are smooth, that is, these estimators are less sensitive to changes in k. All the tail index estimators considered are very unstable for small values of k due to the small number of exceedances. A specific tail index estimate can be obtained from the plots of WLS and RWLS for both datasets.Figure 9
Tail index estimates as a function ofk: (a) SOA data; (b) GH claims.
(a)(b)


## 5. Conclusion

In this paper, we proposed tail index estimators for the Pareto-type of distributions using the regression model. In addition to the ordinary least squares and the ridge regression estimators, we proposed the regularised weighted least squares and the weighted least squares estimators as alternative regression-based reduced-bias estimators. The tail index estimates by the proposed estimators are generally stable and smooth across a broader path ofk. The characteristics of the proposed estimators are as follows:(i)
They are asymptotically consistent, asymptotically unbiased, and asymptotically normally distributed with mean 0 and varianceγ2.(ii)
The MSE curves are low and flat over the central part ofk.(iii)
The plots of their tail index estimates are more stable, smooth, and near horizontal than the Hill, ordinary least squares, and the bias-corrected Hill estimators.In conclusion, comparatively, the proposed estimators are competitive to the existing estimators and can be considered as appropriate estimators of the tail index in terms of MSE, bias, and in real-life application.


## 6. Proofs

Proof of Lemma 1.
(i)
From equation (20) and (22), we have(46)S1θ=∑j=1kW˜jCj=2αk+12αk+1∫011−Eθαkuu−ρdu+o1=22αk/k−ραk/k+1/k2αk/k+1/k1−ρ2−ρ+o1.It follows that(47)limk⟶∞22αk/k−ραk/k+1/k2αk/k+1/k1−ρ2−ρ+o1=22Δ−ρΔ2Δ1−ρ2−ρ=11−ρ,whereΔ=limk⟶∞αk/k and hence, as k⟶∞, we have S1θ⟶a.s1/1−ρ.(ii)
From equation (23),(48)S2θ=∑j=1kW˜jCj2−∑j=1kW˜jCj2=∑j=1kW˜jCj2−11−ρ2.Using equation (20), the first term can be written as(49)∑j=1kW˜jCj2=2αk+12αk+1∫011−Eθαkuu−2ρdu+o1=2αk/k−2ραk/k+1/k2αk/k+1/k1−ρ1−2ρ+o1.It follows that(50)limk⟶∞2αk/k−2ραk/k+1/k2αk/k+1/k1−ρ1−2ρ+o1=2Δ−2ρΔ2Δ1−ρ1−2ρ=11−2ρ,whereΔ=limk⟶∞αk/k. Therefore,(51)S2θ⟶11−2ρ−11−ρ2=ρ21−2ρ1−ρ2,ask⟶∞. That is, S2θ⟶a.sρ2/1−2ρ1−ρ2 as k⟶∞.(iii)
The expressionS˙θ=∑j=1kW˜j2S1θ−Cj can also be written as(52)S˙θ=4kαk+12αk+121k∑j=1k1−θjαkjk+12S1θ−jk+1−ρ=4kαk/k+1/k2αk/k+1/k2∫011−Eθαku211−ρ−u−ρdu+o1.Therefore, ask⟶∞, S˙θ⟶a.s0.(iv)
S¨θ=∑j=1kW˜j2S1θ−Cj2 can also be expressed as(53)S¨θ=4kαk+12αk+121k∑j=1k1−θjαkjk+12S1θ−jk+1−ρ2=4kαk/k+1/k2αk/k+1/k2∫011−Eθαku211−ρ−u−ρ2du+o1.It also follows that, ask⟶∞, S¨θ⟶a.s0.Proof of Lemma 2.
The ProofProof 2 easily follows by using Lemma 1.Proof of Lemma 3.
We observe that(54)−kρk1+ω+ρ+nρ≤Mλθ,Cj≤kρCj2/S2θk1+ω+ρ+nρ.
Therefore, we haveMλθ,Cj⟶o1/k1+ω,0<ω≤0.1, as k⟶∞, which completes the Proof Proof 3..Proof of Lemma 4.
The proof requires the use of large deviation principles (LDP). From equations (15) and (33), b^n,kθ,ρ^=∑j=1kMλθ,CjTj. Given ρ^=ρ, Tj is exponentially distributed with mean(55)μjθ,ρ=ETj|ρ^=ρ,θ=γ+b^n,kρ^,θCjρ^.
Therefore, we have(56)MTk,θ,ρt=EetTk|θ,ρ^=ρ=11−μkθ,ρt.
Using equation (36) and similar calculations as in [32], the moment generating function of b^n,k given the law of θ,ρ^=ρ is(57)Mb^n,kρt=∏j=1k11−μjθ,ρMλθt.
It follows that(58)Mb^n,kkt=∏j=1k11−kμjMλθ,Cjt.
Now using the bound onMλθ,Cj, see Lemma 4 and the Squeeze Theorem, we obtain(59)limk⟶∞1klogMb^n,kkt=0.
Hence, by the Ga¨ rtner Ellis Theorem, conditional on θ,ρ, the statistics b^n,kθ,ρ follows a Large Deviation Principle (LDP) with speed k and a rate function Ix defined as(60)Ix=supη∈ℝηx−0=supη∈ℝηx=0ifx=0,∞ifx≠0,where x∈0,∞. This implies for every ε>0, we have(61)ℙkb^k,n>ε=ℙb^n,k>εk|θ^,ρ^=θ,ρℙθ^,ρ^=θ,ρ=ℙb^n,k∈Γkℙθ^,ρ^=θ,ρ≤e−infx∈ΓkkIx+ok,
whereΓk=ε/k,∞. The typical behaviour of the rate function is when x≠0, therefore(62)limk⟶∞ℙkb^n,k>ε≤0.
Thus,kb^n,k⟶p0 as k⟶∞ and this ends the proof.Proof of Lemma 5.
We observe thatT1,T2,T3,… are independent but not identical distributed random variables.(i)
(63)ETj=EETj|θ,ρ^=Eγ+b^n,kθ,ρ^Cjρ^≤γ+Eb^n,kθ,ρ^≤γ+E1−2ρ^1−ρ^2kρ^+1∑j=1kCjρ^Zj/kρ^2k1+ω+ρ^+nρ^+o1≤γ+E1−2ρ^1−ρ^2ρ^2kωE∑j=1kCjρ^Zj/k+o1a.s.≤γ+E1−2ρ^1−ρ^2ρ^2EγkHkω+o1a.s.≤γ+o1/kω+o1,where0<ω<0.1 and γkH is the Hill estimator. Hence, we have(64)μj=EEθW˜θjTj≤EθW˜jθjγ+o1/kω+o1.(ii)
Letf be the probability density function of Tj given θ,ρ^ and we observe that(65)EZj−μj2+δ|θ,ρ^=ρ=∫0∞Zj−μj2+δfzjdzj=−∫0μjZj−μj2+δfzjdzj+∫μj∞Zj−μj2+δfzjdzj=e−1μj−μj3+δ7+2δ3+δ4+δ+μj3+δ2+δ!1+o1=μj2+δηδ1+o1,whereηδ=1/e−13+δ7+2δ/3+δ4+δ+2+δ!<∞ for 0<δ<∞. Therefore, we have that(66)EZj−μj2+δ=EETj−μj2+δ|θ,ρ^≤ηδ1+o1μj2+δ.
Now defineZj=EθW˜jθjTj−μj,1≤j≤k, and note that(67)Sn2=Var∑j=1kEθZj=∑j=1kVarEθW˜jθjTj=∑j=1kEθW˜jθ2VarTj=γ2∑j=1kEθW˜jθ2.
Hence,(68)limk⟶∞1Sn2+δ∑j=1kEZj−μj2+δ≤limk⟶∞ηδ1+o1γ+o1/kω+o12+δ∑j=1kEθW˜jθj2+δγ2∑j=1kEθW˜jθ21+δ/2=ηδlimk⟶∞∑j=1kEθW˜jθj2+δ∑j=1kEθW˜jθ21+δ/2≤ηδlimk⟶∞21+δ/2k−δ/2καk1+δ/21+ok1−ok1+δ/2=0.Proof of Theorem 1.
Using Lemma5 and Lemma 7, we can prove Theorem 3. It has been established in Lemma 5 that kb^n,k⟶p0 as k⟶∞. Lemma 7 also establishes that the Lyapunov’s condition holds for Central Limit Theorem; hence, by the Lyapunov’s Central Limit Theorem,(69)kEθγ^RWλ,θ−γ⟶dNμ,σ2,as k⟶∞.
Therefore, all we need to complete the ProofProof 6, is to specify the parameters of the normal distribution.
Recall from (16) that(70)γ^kλ≔Eθγ^RWλ,θ=Eθ∑j=1kW^jTj−b^n,k∑j=1kW^jCj.
Also, from Lemma5, asymptotically, the second term on the right hand side vanishes, so we would concentrate on the first term of the expression only. Let(71)Sk=kγ^kλ−γ=kEθ∑j=1kW˜jTj−b^n,k∑j=1kW˜jCj−γ.
The expected value ofSk is given as(72)ESk=Eθk∑j=1kW˜jETj−kb^n,k∑j=1kW˜jCj−kγ=EθEρk∑j=1kW˜jγ+b^n,kρCjρ−kb^n,kρ∑j=1kW˜jCjρ−kγ=EθEρ0=0.
Hence,ESk⟶μ=0 as k⟶∞. Recall that k⟶∞, kb^n,k⟶p0, 2καk⟶1 and by assumption θj and Tj are independent, therefore, the variance of Sk is given by(73)VarSk=kVarEθ∑j=1kW^jTj=4kκ2αk∑j=1kVarEθ1−θjαkjk+1Tj=4kκ2αk∑j=1kEθ1−θjαkjk+12VarTj=4kκ2αk∑j=1k1−1αk+1jk+12VarTj=2καk2γ21−1αk+1+okk=γ21−1αk+1+okk+o1.
Using the assumptionαk/k⟶Δ<∞, as k⟶∞, we have VarSk⟶σ2=γ2, which completes the Proof of Theorem 1.



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*Source: 5064875-2022-10-27.xml*