# POT-Based Estimation of the Renewal Function of Interoccurrence Times of Heavy-Tailed Risks

Authors: Abdelhakim Necir; Abdelaziz Rassoul; Djamel Meraghni
Journal: Journal of Probability and Statistics
 (2010)

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## Abstract

Making use of the peaks over threshold (POT) estimation method,
we propose a semiparametric estimator for the renewal function of
interoccurrence times of heavy-tailed insurance claims with infinite
variance. We prove that the proposed estimator is consistent and
asymptotically normal, and we carry out a simulation study to compare its finite-sample behavior with respect to the nonparametric one.
Our results provide actuaries with confidence bounds for the renewal
function of dangerous risks.


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## Body

## 1. Introduction

LetX1,X2,… be independent and identically distributed (iid) positive random variables (rvs), representing claim interoccurrence times of an insurance risk, with common distribution function (df) F having finite mean μ and variance σ2. Let(1.1)𝕊m:={X1+⋯+Xm,m=1,2,…,0,m=0
be the claim occurrence times, and define the number of claims recorded over the time interval [0,t] by(1.2)ℕ(t):=max{m≥0,𝕊m≤t}.
The corresponding renewal function is defined by(1.3)ℝ(t):=E[ℕ(t)]=∑k=1∞F(k)(t),t>0,
where F(k) is the k-fold convolution of F for k≥1.The renewal theory has proved to be a powerful tool in stochastic modeling in a wide variety of applications such as reliability theory, where a renewal process is used to model the successive repairs of a failed machine (see [1]), risk theory, where a renewal process is used to model the successive occurrences of risks (see [2, 3]), inventory theory, where a renewal process is used to model the successive times between demand points (see [4]), manpower planning, where a renewal process is used to model the sequence of resignations from a given job (see [5]), and warranty analysis, where a renewal process is used to model the successive purchases of a new item following the expiry of a free-replacement warranty (see [6]). Therefore, the need for renewal function estimates seems more than pressing in many practical problems. For a summary of renewal theory, one refers to Feller [7], Asmussen [8], and Resnick [9].Statistical estimation of the renewal function has been considered in several ways. Using a nonparametric approach, Frees [10] introduced two estimators based on the empirical counterparts of F and F(k) by suitably truncating the sum in (1.3). Zhao and Subba Rao [11] proposed an estimation method based on the kernel estimate of the density and the renewal equation. A histogram-type estimator, resembling to the second estimator of Frees, was given by Markovich and Krieger [12].WhenE[X2]=∞, Sgibnev [13] gave an asymptotic approximation of (1.3) as follows:(1.4)ℝ(t)-tμ~1E[X2]∫0t(∫y∞F¯(x)dx)dy,
with F¯:=1-F being the tail of F.By replacingF by its empirical counterpart Fn in (1.4), Bebbington et al. [14] recently proposed a nonparametric estimator for ℝ(t) in the case where F is of infinite variance, given by(1.5)ℝ̃n(t):=tμ̃+1μ̃2∫0t(∫y∞F¯n(x)dx)dy,
where μ̃ and μ̃2, respectively, represent the first and second sample moments of F. Their main result says that whenever F belongs to the domain of attraction of a stable law Sα with 1/2<α<1 (see, e.g., [15]), the df of ℝ̃n(t) converges, for suitable normalizing constants, to Sα. This result provides confidence bounds for ℝ(t) with respect to the quantiles of Sα.In general, people prefer estimators having simple formulas and carrying some kind of asymptotic normality property in order to facilitate confidence interval construction. From this point of view, the estimatorℝ̃n(t) may not be as satisfactory to the users as it should be. Then an alternative estimator to ℝ̃n(t) would be more useful in practice. Our task is to use the extreme value theory tools to construct such an alternative estimator.Indeed, an important class of models having infinite second-order moments is the set of heavy-tailed distributions (e.g., Pareto, Burr, Student, etc.). A dfF is said to be heavy-tailed with tail index ξ>0 if(1.6)F¯(x)=cx-1/ξ(1+x-δ𝕃(x)),asx→∞,
for ξ∈(0,1),δ>0, and some real constant c, with 𝕃 a slowly varying function at infinity, that is, 𝕃(tx)/𝕃(x)→1 as x→∞ for any t>0. For details on these functions, see Chapter 0 in Resnick [16] or Seneta [17]. Notice that when ξ∈(1/2,1) we have μ<∞ and E[X2]=∞. In this case, an asymptotic approximation of the renewal function ℝ(t) is given in (1.4).Prior to Sgibnev [13], Teugels [18] obtained an approximation of ℝ(t) when F is heavy-tailed with tail index ξ∈(1/2,1):(1.7)ℝ(t)-tμ~ξ2t2F¯(t)μ2(1-ξ)(2ξ-1),ast→∞.
Extreme value theory allows for an accurate modeling of the tails of any unknown distribution, making the (semiparametric) statistical inference more accurate for heavy-tailed distributions. Indeed, the semiparametric approach permits extrapolating beyond the largest value of a given sample while the nonparametric one does not since the empirical df vanishes outside the sample. This represents a big handicap for those dealing with heavy-tailed data.Extreme value theory has two aspects. The first one consists in approximating the tail distribution by the generalized extreme value (GEV) distribution, thanks to Fisher-Tippett theorem (see [19, 20]). The second aspect (commonly known as POT method) is based on Balkema-de Haan result which says that the distribution of the excesses over a fixed threshold is approximated by the generalized Pareto distribution (GPD) (see [21, 22]). Those interested in extreme value theory and its applications are referred to the textbooks of de Haan and Ferriera [23] and Embrechts et al. [24]. In our situation, we have a fixed threshold equal to the horizon t=tn (see Section 3). Therefore, the POT method would be the appropriate choice to derive an estimator for ℝ(t) by exploiting the heavy-tail property of df F used in approximation (1.4). The asymptotic normality of our estimator is established under suitable assumptions.The remainder of the paper is organized as follows. In Section2, we introduce the GPD approximation, mostly known as the POT method. A new estimator of the renewal function ℝ(t) is proposed in Section 3, along with two main results on its limiting behavior. Section 4 is devoted to a simulation study. The proofs are postponed until Section 5.


## 2. GPD Approximation

The distribution of the excesses, over a “fixed” thresholdt, pertaining to df F is defined by(2.1)Ft(y):=P(X1-t≤y∣X1>t),fory>0.
It is shown, in Balkema and de Haan [21] and Pickands [22], that Ft is approximated by a generalized Pareto distribution (GPD) function 𝔾ξ,β with shape parameter ξ∈ℝ and scale parameter β=β(t)>0, in the following sense:(2.2)supy>0|Ft(y)-𝔾ξ,β(y)|=O(t-δ𝕃(t)),ast→∞,
where t-δ𝕃(t)→0 as t→∞ for any δ>0. The GPD function 𝔾ξ,β is a two-parameter df defined by(2.3)𝔾ξ,β(y)={1-(1+ξyβ)-1/ξ,ξ≠0,1-exp(-yβ),ξ=0,
for 0≤y<∞ if ξ≥0 and 0≤y<-β/ξ if ξ<0.LetY1,…,YN be iid rvs with exact GPD 𝔾ξ,β. It is well known by standard arguments (see, e.g., [25, Chapter 9]) that there exists, with probability 1 as N tends to infinity, a local maximum (ξ̂N,β̂N) for the Log-Likelihood of 𝔾ξ,β's density based on the sample (Y1,…,YN). In this case, by Theorem 3.7 page 447 in the work of Lehmann and Casella [26], we infer that ξ̂N and β̂N are consistent estimators of ξ and β. Moreover, these estimators are asymptotically normal provided that ξ>-1/2. The extension to ξ≤-1/2 was investigated by Smith [27].Suppose now thatY1,…,YN are drawn not from 𝔾ξ,β, but from Ft. In view of the asymptotic approximation (2.2), Smith [27] has proposed estimates for (ξ,β) via the Maximum Likelihood approach. The obtained estimators (ξ̂N,β̂N) are solutions of the following system:(2.4)1N∑i=1Nlog(1+ξyiβ)=ξ,1N∑i=1Nyi/β1+yi/β=11+ξ,
where (y1,…,yN) is a realization of (Y1,…,YN).Lettingt=tN→∞ asN→∞ and βN=tNξ and making use of (2.2), Smith [28] established, in Theorem 3.2, the asymptotic normality of (ξ̂N,β̂N) as follows:(2.5)N(β̂NβN-1ξ̂N-ξ)→𝒟𝒩2(0,ℚ-1)asN→∞,
where(2.6)ℚ-1=(1+ξ)(2-1-11+ξ),
provided that NtN-δ𝕃(tN)→0 as N→∞ and x↦x-δ𝕃(x) is nonincreasing near infinity. In the case NtN-δ𝕃(tN)↛0, the limiting distribution in (2.5) is biased. Here →𝒟 denotes convergence in distribution and 𝒩2(ω,Σ) stands for the bivariate normal distribution with mean vector ω and covariance matrix Σ.


## 3. Estimating the Renewal Function in Infinite Time

Since we are interested in the renewal function in infinite time, we must assume that timet is large enough and for asymptotic considerations, we will assume that t depends on the sample size n. That is, t=tn, with tn→∞ as n→∞. Relation (1.7) suggests that in order to construct an estimator of ℝ(tn), we need to estimate μ,ξ and F¯(tn). Let n=n(t) be the number of Xis, which are observed on horizon tn and denoted by(3.1)Ntn:=card({Xi>tn:1≤i≤n}),
the number of exceedances over tn, with card(K) being the cardinality of set K. Notice that Ntn is a binomial rv with parameters n and pn:=F¯(tn) for which the natural estimator is p̂n:=Nt/n.Select, from the sample(X1,…,Xn), only those observations Xi1,…,XiNtn that exceed tn. The Nt excesses(3.2)Ej:n:=Xij-tn,j=1,…,Ntn
are iid rvs with common df Ftn. As seen in Section 2, the maximum likelihood estimators (ξ̂n,β̂n) are solutions of the following system:(3.3)1vn∑j1vnlog(1+ξej:nβ)=ξ,1v∑j=1vnej:n/β1+ej:n/β=11+ξ,
where vn is an observation of Ntn and the vector (e1:n,…,evn:n) a realization of (E1:n,…,ENtn:n). Regarding the distribution mean μ=E[X1], we know that, for ξ∈(0,1/2],X1 has finite variance and therefore μ could naturally be estimated by the sample mean X¯:=n-1Sn which, by the Central Limit Theorem (CLT), is asymptotically normal. Whereas for ξ∈(1/2,1),X1 has infinite variance, in which case the CLT is no longer valid. This case is frequently met in real insurance data (see, e.g., [29]). Using the GPD approximation, Johansson [30] has proposed an alternative estimator for μ=∫0∞xdF(x). For each n≥1, we write μ as the sum of two components:(3.4)μn*:=∫0tnxdF(x),τn:=∫tn∞xdF(x)=-∫0∞(tn+s)dF¯(tn+s).
Johansson [30] defined his estimator of μ, by estimating both F(x) and F¯(tn+s), as follows:(3.5)μ̂n(J):=∫0tnxdFn(x)-∫0∞(tn+s)dF¯̂(tn+s),
where Fn is the empirical df based on the sample (X1,…,Xn) and F¯̂(tn+s) is an estimate of F¯(tn+s) obtained from the relation(3.6)F¯tn(s)=F¯(tn+s)F¯(tn),s>0,
which implies that F¯(tn+s)=pnF¯tn(s),s>0. Approximation (2.2) motivates us to estimate F¯tn(s) by F¯̂tn(s):=𝔾¯ξ̂n,β̂n(s),s>0. Hence, an estimate of F¯(tn+s) is(3.7)F¯̂(tn+s):=p̂n𝔾¯ξ̂n,β̂n(s),s>0.
By integrating (3.5), we get(3.8)μ̂n(J)=1n∑i=1nXi1{Xi≤tn}+p̂n(tn+β̂n1-ξ̂n)=:μ̂n*+τ̂n,
with ξ̂n∈(0,1) with large probability. Here, 1K denotes the indicator function of set K. Respectively, substituting μ̂n(J),ξ̂n, and p̂n for μ,ξ and F¯(tn) in (1.7) yields the following estimator for the renewal function ℝ(tn)(3.9)ℝ̂n(tn):=tnμ̂n(J)+ξ̂n2tn2p̂nμ̂n(J)2(1-ξ̂n)(2ξ̂n-1).
The asymptotic behavior of ℝ̂n(tn) is given by the following two theorems.Theorem 3.1.
LetF be a df fulfilling (1.6) with ξ∈(1/2,1). Suppose that 𝕃 is locally bounded in [x0,+∞) for x0≥0 and x↦x-δ𝕃(x) is nonincreasing near infinity, for some δ>0. Then, for any tn=O(nαξ/4) with α∈(0,1), one has
(3.10)ℝ̂n(tn)-ℝ(tn)=Oℙ(n(α/2)(ξ-1/4)-1/2),asn→∞.Theorem 3.2.
LetF be as in Theorem 3.1. Then for any tn=O(nαξ/4) with α∈(4/(1+2ξδ),1), we have
(3.11)nsntn(ℝ̂n(tn)-ℝ(tn))→𝒟𝒩(0,1),asn→∞,
where
(3.12)sn2:=θ12+pn(1-pn)γn2(θ2+θ1(tn+βn1-ξ))2+pnγn2(θ3+θ1pnβn(1-ξ)2)2+θ12βn2pn3γn2(1-ξ)2-θ1βnpn2γn2(θ3+θ1pnβn(1-ξ)2)
with
(3.13)θ1:=-1μ2-2ξ2tnpnμ3(1-ξ)(2ξ-1),θ2:=ξ2tnμ2(1-ξ)(2ξ-1),θ3:=tnpnμ2(1-ξ)(2ξ-1)(2ξ+4ξ3-3ξ2(1-ξ)(2ξ-1)),pn:=F¯(tn),βn:=tnξ, and γn2:=Var(X11{X1≤tn}).


## 4. Simulation Study

In this section, we carry out a simulation study (by means of the statistical softwareR, see [31]) to illustrate the performance of our estimation procedure, through its application to sets of samples taken from two distinct Pareto distributions F(x)=1-x-1/ξ,x>1 (with tail indices ξ=3/4 and ξ=2/3). We fix the threshold at 4, which is a value above the intermediate statistic corresponding to the optimal fraction of upper-order statistics in each sample. The latter is obtained by applying the algorithm of Cheng and Peng [32]. For each sample size, we generate 200 independent replicates. Our overall results are then taken as the empirical means of the values in the 200 repetitions.A comparison with the nonparametric estimator is done as well. In the graphical illustration, we plot both estimators versus the sample size ranging from1000 to 20000.Figure1 clearly shows that the new estimator is consistent and that it is always better than the nonparametric one. For the numerical investigation, we take samples of sizes 1000,2000 and 5000. In each case, we compute the semiparametric estimate R̂ as well as the nonparametric estimate R̃. We also provide the bias and the root mean squared error (rmse).Plots of the new and sample estimators of the renewal function, of interoccurrence times of Pareto-distributed claims with tail indices 2/3 (a) and 3/4 (b), versus the sample size. The horizontal line represents the true value of the renewal functionℝ(t) evaluated at t=4.
(a)(b)The results are summarized in Tables1 and 2 for ξ=3/4 and ξ=2/3 respectively. We notice that, regardless of the tail index value and the sample size, the semiparametric estimation procedure is more accurate than the nonparametric one.Table 1
Semiparametric and nonparametric estimates of the renewal function of interoccurrence times of Pareto-distributed claims with shape parameter 3/4. Simulations are repeated 200 times for different sample sizes.
True valueR=1.708Sample sizeSemiparametricR̂NonparametricR̃MeanBiasRMSEMeanBiasRMSE10001.696-0.0130.2502.1410.4330.55320001.7190.0110.1831.9080.1990.28850001.705-0.0030.1191.686-0.0220.168Table 2
Semiparametric and nonparametric estimates of the renewal function of interoccurence times of Pareto-distributed claims with shape parameter 2/3. Simulations are repeated 200 times for different sample sizes.
True valueR=2.222Sample sizeSemiparametricR̂NonparametricR̃MeanBiasRMSEMeanBiasRMSE10002.2650.0420.1852.4160.1930.22920002.2470.0240.1572.054-0.1670.22350002.2230.0010.1292.073-0.1490.192


## 5. Proofs

The following tools will be instrumental for our needs.Proposition 5.1.
LetF be a df fulfilling (1.6) with ξ∈(1/2,1),δ>0, and some real c. Suppose that 𝕃 is locally bounded in [x0,+∞) for x0≥0. Then for n large enough and for any tn=O(nαξ/4),α∈(0,1), one has
(5.1)pn=c(1+o(1))n-α/4,γn2=O(n(α/2)(ξ-1/2)),sn2=O(n(α/2)(ξ-1/2)),npntn-δ𝕃(tn)=O(n-α/8-αξδ/4+1/2),
where pn,γn2, and sn2 are those defined in Theorem 3.2.Lemma 5.2.
Under the assumptions of Theorem3.2, one has, for any real numbers u1,u2,u3 and u4,(5.2)E[exp{iu1nγn(μ̂n*-μn*)+inpn(u2,u3)(β̂nβn-1ξ̂n-ξ)+iu4n(p̂n-pn)pn(1-pn)}]→exp{-u122-12(u2,u3)ℚ-1(u2u3)-u422},asn→∞,
where i2=-1.Proof.
We will only prove the second result, the other ones are straightforward from (1.6). Let x0>0 be such that F¯(x)=cx-1/ξ(1+x-δ𝕃(x)), for x>x0. Then for n large enough, we have
(5.3)E[X11{X1≤tn}]=∫0tnxdF(x)=∫0x0xdF(x)+∫x0tnxdF(x).
Recall that μ<∞, hence ∫0x0xdF(x)<∞. Making use of the proposition assumptions, we get E[X11{X1≤tn}]=O(1) and E[X121{X1≤tn}]=O(tn2-1/ξ) and therefore γn2=O(nα/2(ξ/2-1)).Proof of Lemma5.2.
See Johansson [30].Proof of Theorem3.1.
We may readily check that for all largen,(5.4)(ℝ̂n(tn)-ℝn(tn))tn~An+Bn+Cn,
where
(5.5)An:=(-1μ̂n(J)μ-ξ2tnpn(μ̂n(J)+μ)μ̂n(J)2μ2(1-ξ)(2ξ-1))(μ̂n(J)-μ),Bn:=ξ̂n2tnμ̂n(J)2(1-ξ̂n)(2ξ̂n-1)(p̂n-pn),Cn:=tnpnμ̂n(J)2(1-ξ̂n)(2ξ̂n-1)×(ξ̂n+ξ+2ξ2(ξ̂n+ξ)-3ξ2(1-ξ)(2ξ-1))(ξ̂n-ξ).
Johansson [30] proved that there exists a bounded sequence kn such that
(5.6)μ̂n(J)-μ=Oℙ(γnknn),
hence μ̂n(J)-μ=Oℙ(n(α/4)(ξ-1/2)-1/2). The first result of the proposition yields that
(5.7)tnpn(μ̂n(J)-μ)=Oℙ(n(α/4)(2ξ-3/2)-1/2).
Since (α/4)(2ξ-3/2)-1/2<0, then tnpn(μ̂n(J)-μ)=oℙ(1). On the other hand, by the CLT we have
(5.8)p̂n-pn=Oℙ(pnn),
then tn(p̂n-pn)=Oℙ(n(α/4)(ξ-1/2)-1/2)=oℙ(1). On the other hand, Smith [28], yields(5.9)ξ̂n-ξ=Oℙtn-δ𝕃(tn),
it follows that,  ξ̂n2tn(p̂n-pn)=Oℙ(n(α/4)(ξ(1-2δ)-1/2)-1/2)=oℙ(1), therefore
(5.10)ξ̂n2tn(p̂n-pn)μ̂n(J)2(1-ξ̂n)(2ξ̂n-1)=oℙ(1),tnpn(ξ̂n-ξ)=Oℙ(n(α/4)(ξ(1-δ)-1))=oℙ(1),ξ̂ntnpn(ξ̂n-ξ)=Oℙ(n(α/4)(ξ(1-2δ)-1))=oℙ(1),pn(ξ̂n-ξ)=Oℙ(n(-α/4)((1+ξδ)))=oℙ(1).
Thus,
(5.11)(ξ̂n+ξ)μ̂n(J)2(1-ξ̂n)(2ξ̂n-1)tnpn(ξ̂n-ξ)→ℙ0,tnpn(2ξ2(ξ̂n+ξ)-3ξ2)μ̂n(J)2(1-ξ̂n)(2ξ̂n-1)(1-ξ)(2ξ-1)(ξ̂n-ξ)→ℙ0asn→∞.
Therefore for all large n, we get ℝ̂(tn)-ℝ(tn)=Oℙ(n(α/2)(ξ-1/4)-1/2), as sought.Proof of Theorem3.2.
From the proof of Theorem3.1, for all large n, it is easy to verify that
(5.12)(ℝ̂n(tn)-ℝn(tn))tn=θ1(1+oℙ(1))(μ̂n(J)-μ)+θ2(1+oℙ(1))(p̂n-pn)+θ3(1+oℙ(1))(ξ̂n-ξ),
where
(5.13)θ1=-1μ2-2ξ2tnpnμ3(1-ξ)(2ξ-1),θ2=ξ2tnμ2(1-ξ)(2ξ-1),θ3=tnpnμ2(1-ξ)(2ξ-1)(2ξ+4ξ3-3ξ2(1-ξ)(2ξ-1)).
Multiplying by n/γn and using the proposition and the lemma together with the continuous mapping theorem, we find that
(5.14)nγntn(ℝ̂n(tn)-ℝn(tn))=θ1(1+oℙ(1))nγn(μ̂n(J)-μ)+θ2(1+oℙ(1))nγn(p̂n-pn)+θ3(1+oℙ(1))nγn(ξ̂n-ξ).
On the other hand, from Johansson [30], we have for all large n(5.15)nγn(μ̂n(J)-μ)=nγn(μ̂n*-μn*)+(tn+βn1-ξn)nγn(p̂n-pn)+pnβn(1-ξ)2nγn(ξ̂n-ξ)+pn1-ξnγn(β̂n-βn)+oℙ(1).
This enables us to rewrite (n/γntn)(ℝ̂n(tn)-ℝn(tn)) into
(5.16)θ1nγn(μ̂n*-μn*)+pn(1-pn)γn(θ2+θ1(tn+βn1-ξ))n(p̂n-pn)pn(1-pn)+θ1βnpnpnγn(1-ξ)npn(β̂nβn-1)+pnγn(θ3+θ1pnβn(1-ξ)2)npn(ξ̂n-ξ)+oℙ(1),ℚ-1=(1+ξ)(2-1-11+ξ).
In view of Lemma 5.2, we infer that for all large n, the previous quantity is
(5.17)θ1𝒲1+pn(1-pn)γn(θ2+θ1(tn+βn1-ξ))𝒲2+2(1+ξ)θ1βnpnpnγn(1-ξ)𝒲3+(1+ξ)pnγn(θ3+θ1pnβn(1-ξ)2)𝒲4+oℙ(1),
where (𝒲i)i=1,4 are standard normal rvs with E[WiWj]=0 for every i,j=1,…,4 with i≠j, except for
(5.18)E[W3W4]=E[2(1+ξ)npn(β̂nβn-1)(1+ξ)npn(ξ̂n-ξ)]=2(1+ξ)(1+ξ)E[npn(β̂nβn-1)npn(ξ̂n-ξ)]=-2(1+ξ)(1+ξ)2.
Therefore, the rv (n/γntn)(ℝ̂n(tn)-ℝn(tn)) is Gaussian with mean zero with asymptotic variance
(5.19)Kn2:=θ12+pn(1-pn)γn2(θ2+θ1(tn+βn1-ξ))2+2(1+ξ)θ12βn2pn3γn2(1-ξ)2+(1+ξ)2pnγn2(θ3+θ1pnβn(1-ξ)2)2-2θ1βnpn2(1+ξ)4(1-ξ)γn2(θ3+θ1pnβn(1-ξ)2)+oℙ(1).
Observe now that Kn2=sn2+oℙ(1), where sn2 is that in (3.12), this completes the proof of Theorem 3.2.


## 6. Conclusion

In this paper, we have proposed a new estimator for the renewal function of heavy-tailed claim interoccurence times, via a semiparametric approach. Our considerations are based on one aspect of the extreme value theory, namely, the POT method. We have proved that our estimator is consistent and asymptotically normal. Moreover, simulations show that it is more accurate than the nonparametric estimator given by Bebbington et al. [14].



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