About this Journal Submit a Manuscript Table of Contents
Journal of Probability and Statistics
Volume 2010 (2010), Article ID 965672, 14 pages
http://dx.doi.org/10.1155/2010/965672
Research Article

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

1Laboratory of Applied Mathematics, University Mohamed Khider, P.O. Box 145, Biskra 07000, Algeria
2Ecole Nationale Superieure d'Hydraulique, Guerouaou, BP 31, Blida 09000, Algeria

Received 13 October 2009; Accepted 24 February 2010

Academic Editor: Ričardas Zitikis

Copyright © 2010 Abdelhakim Necir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let be independent and identically distributed (iid) positive random variables (rvs), representing claim interoccurrence times of an insurance risk, with common distribution function (df) having finite mean and variance Let be the claim occurrence times, and define the number of claims recorded over the time interval by The corresponding renewal function is defined by where is the -fold convolution of for

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 and 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].

When Sgibnev [13] gave an asymptotic approximation of (1.3) as follows: with being the tail of

By replacing by its empirical counterpart in (1.4) Bebbington et al. [14] recently proposed a nonparametric estimator for in the case where is of infinite variance, given by where and , respectively, represent the first and second sample moments of Their main result says that whenever belongs to the domain of attraction of a stable law with (see, e.g., [15]), the df of converges, for suitable normalizing constants, to This result provides confidence bounds for with respect to the quantiles of

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 estimator may not be as satisfactory to the users as it should be. Then an alternative estimator to 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 df is said to be heavy-tailed with tail index if for , and some real constant with a slowly varying function at infinity, that is, as for any For details on these functions, see Chapter in Resnick [16] or Seneta [17]. Notice that when we have and In this case, an asymptotic approximation of the renewal function is given in (1.4)

Prior to Sgibnev [13], Teugels [18] obtained an approximation of when is heavy-tailed with tail index : 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 (see Section 3). Therefore, the POT method would be the appropriate choice to derive an estimator for by exploiting the heavy-tail property of df 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 Section 2, we introduce the GPD approximation, mostly known as the POT method. A new estimator of the renewal function 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” threshold pertaining to df is defined by It is shown, in Balkema and de Haan [21] and Pickands [22], that is approximated by a generalized Pareto distribution (GPD) function with shape parameter and scale parameter in the following sense: where as for any The GPD function is a two-parameter df defined by for if and if

Let be iid rvs with exact GPD It is well known by standard arguments (see, e.g., [25, Chapter 9]) that there exists, with probability as tends to infinity, a local maximum for the Log-Likelihood of 's density based on the sample In this case, by Theorem page 447 in the work of Lehmann and Casella [26], we infer that and are consistent estimators of and Moreover, these estimators are asymptotically normal provided that The extension to was investigated by Smith [27].

Suppose now that are drawn not from but from In view of the asymptotic approximation (2.2) Smith [27] has proposed estimates for via the Maximum Likelihood approach. The obtained estimators are solutions of the following system: where is a realization of

Letting as and and making use of (2.2) Smith [28] established, in Theorem , the asymptotic normality of as follows: where provided that as and is nonincreasing near infinity. In the case the limiting distribution in (2.5) is biased. Here denotes convergence in distribution and stands for the bivariate normal distribution with mean vector and covariance matrix

3. Estimating the Renewal Function in Infinite Time

Since we are interested in the renewal function in infinite time, we must assume that time is large enough and for asymptotic considerations, we will assume that depends on the sample size That is, with as Relation (1.7) suggests that in order to construct an estimator of we need to estimate and Let be the number of s, which are observed on horizon and denoted by the number of exceedances over with being the cardinality of set Notice that is a binomial rv with parameters and for which the natural estimator is

Select, from the sample only those observations that exceed The excesses are iid rvs with common df As seen in Section 2, the maximum likelihood estimators are solutions of the following system: where is an observation of and the vector a realization of Regarding the distribution mean we know that, for has finite variance and therefore could naturally be estimated by the sample mean which, by the Central Limit Theorem (CLT), is asymptotically normal. Whereas for 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 For each we write as the sum of two components: Johansson [30] defined his estimator of by estimating both and as follows: where is the empirical df based on the sample and is an estimate of obtained from the relation which implies that Approximation (2.2) motivates us to estimate by Hence, an estimate of is By integrating (3.5), we get with with large probability. Here, denotes the indicator function of set Respectively, substituting , and for and in (1.7) yields the following estimator for the renewal function The asymptotic behavior of is given by the following two theorems.

Theorem 3.1. Let be a df fulfilling (1.6) with Suppose that is locally bounded in for and is nonincreasing near infinity, for some Then, for any with one has

Theorem 3.2. Let be as in Theorem 3.1. Then for any with we have where with , and

4. Simulation Study

In this section, we carry out a simulation study (by means of the statistical software R, see [31]) to illustrate the performance of our estimation procedure, through its application to sets of samples taken from two distinct Pareto distributions (with tail indices and We fix the threshold at , 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 independent replicates. Our overall results are then taken as the empirical means of the values in the repetitions.

A comparison with the nonparametric estimator is done as well. In the graphical illustration, we plot both estimators versus the sample size ranging from to

Figure 1 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 and In each case, we compute the semiparametric estimate as well as the nonparametric estimate We also provide the bias and the root mean squared error (rmse).

fig1
Figure 1: 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 function evaluated at

The results are summarized in Tables 1 and 2 for and 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.

tab1
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.
tab2
Table 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.

5. Proofs

The following tools will be instrumental for our needs.

Proposition 5.1. Let be a df fulfilling (1.6) with , and some real Suppose that is locally bounded in for Then for large enough and for any one has where , and are those defined in Theorem 3.2.

Lemma 5.2. Under the assumptions of Theorem 3.2, one has, for any real numbers and where

Proof. We will only prove the second result, the other ones are straightforward from (1.6). Let be such that for Then for large enough, we have Recall that hence Making use of the proposition assumptions, we get and and therefore

Proof of Lemma 5.2. See Johansson [30].

Proof of Theorem 3.1. We may readily check that for all large where Johansson [30] proved that there exists a bounded sequence such that hence The first result of the proposition yields that Since then On the other hand, by the CLT we have then On the other hand, Smith [28], yields it follows that, therefore Thus, Therefore for all large we get as sought.

Proof of Theorem 3.2. From the proof of Theorem 3.1, for all large it is easy to verify that where Multiplying by and using the proposition and the lemma together with the continuous mapping theorem, we find that On the other hand, from Johansson [30], we have for all large This enables us to rewrite into In view of Lemma 5.2, we infer that for all large the previous quantity is where are standard normal rvs with for every with except for Therefore, the rv is Gaussian with mean zero with asymptotic variance Observe now that where 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].

Acknowledgment

The authors are grateful to the referees whose suggestions led to an improvement of the paper.

References

  1. D. R. Cox, Renewal Theory, Methuen, London, UK, 1962. View at MathSciNet
  2. P. Embrechts, M. Maejima, and E. Omey, “Some limit theorems for generalized renewal measures,” Journal of the London Mathematical Society, vol. 31, no. 1, pp. 184–192, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. P. Embrechts, M. Maejima, and J. Teugels, “Asymptotic behavior of compound distributions,” Astin Bulletin, vol. 15, pp. 45–48, 1985.
  4. S. M. Ross, Stochastic Processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1983. View at MathSciNet
  5. D. J. Bartholomew and A. F. Forbes, Statistical Techniques for Man-Power Planning, Wiley, Chichester, UK, 1979.
  6. W. R. Blischke and E. M. Scheuer, “Applications of renewal theory in analysis of the free-replacement warranty,” Naval Research Logistics Quarterly, vol. 28, no. 2, pp. 193–205, 1981. View at Zentralblatt MATH · View at MathSciNet
  7. W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2, Wiley, New York, NY, USA, 2nd edition, 1971.
  8. S. Asmussen, Applied Probability and Queues, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Chichester, UK, 1987. View at MathSciNet
  9. S. Resnick, Adventures in Stochastic Processes, Birkhäuser, Boston, Mass, USA, 1992. View at MathSciNet
  10. E. W. Frees, “Nonparametric renewal function estimation,” The Annals of Statistics, vol. 14, no. 4, pp. 1366–1378, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Q. Zhao and S. Subba Rao, “Nonparametric renewal function estimation based on estimated densities,” Asia-Pacific Journal of Operational Research, vol. 14, no. 1, pp. 115–126, 1997. View at Zentralblatt MATH · View at MathSciNet
  12. N. M. Markovich and U. R. Krieger, “Nonparametric estimation of the renewal function by empirical data,” Stochastic Models, vol. 22, no. 2, pp. 175–199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. S. Sgibnev, “On the renewal theorem in the case of infinite variance,” Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 22, no. 5, pp. 178–189, 1981. View at MathSciNet
  14. M. Bebbington, Y. Davydov, and R. Zitikis, “Estimating the renewal function when the second moment is infinite,” Stochastic Models, vol. 23, no. 1, pp. 27–48, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. V. M. Zolotarev, One-Dimensional Stable Distributions, vol. 65 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1986. View at MathSciNet
  16. S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, vol. 4 of Applied Probability. A Series of the Applied Probability Trust, Springer, New York, NY, USA, 1987. View at MathSciNet
  17. E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508, Springer, Berlin, Germany, 1976. View at MathSciNet
  18. J. L. Teugels, “Renewal theorems when the first or the second moment is infinite,” Annals of Mathematical Statistics, vol. 39, pp. 1210–1219, 1968. View at Zentralblatt MATH · View at MathSciNet
  19. R. A. Fisher and L. H. C. Tippett, “Limiting forms of the frequency distribution of the largest or smallest member of a sample,” Proceedings of the Cambridge Philosophical Society, vol. 24, pp. 180–190, 1928.
  20. B. Gnedenko, “Sur la distribution limite du terme maximum d'une série aléatoire,” Annals of Mathematics, vol. 44, pp. 423–453, 1943. View at Zentralblatt MATH · View at MathSciNet
  21. A. A. Balkema and L. de Haan, “Residual life time at great age,” Annals of Probability, vol. 2, pp. 792–804, 1974. View at Zentralblatt MATH · View at MathSciNet
  22. J. Pickands III, “Statistical inference using extreme order statistics,” The Annals of Statistics, vol. 3, pp. 119–131, 1975. View at Zentralblatt MATH · View at MathSciNet
  23. L. de Haan and A. Ferreira, Extreme Value Theory: An Introduction, Springer Series in Operations Research and Financial Engineering, Springer, New York, NY, USA, 2006. View at MathSciNet
  24. P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance, vol. 33 of Applications of Mathematics, Springer, Berlin, Germany, 1997. View at MathSciNet
  25. D. R. Cox and D. V. Hinkley, Theoretical Statistics, Chapman and Hall, London, UK, 1974. View at MathSciNet
  26. E. L. Lehmann and G. Casella, Theory of Point Estimation, Springer Texts in Statistics, Springer, New York, NY, USA, 2nd edition, 1998. View at MathSciNet
  27. R. L. Smith, “Maximum likelihood estimation in a class of nonregular cases,” Biometrika, vol. 72, no. 1, pp. 67–90, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. R. L. Smith, “Estimating tails of probability distributions,” The Annals of Statistics, vol. 15, no. 3, pp. 1174–1207, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Beierlant, G. Matthys, and G. Dierckx, “Heavy-tailed distributions and rating,” Astin Bulletin, vol. 31, no. 1, pp. 37–58, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  30. J. Johansson, “Estimating the mean of heavy-tailed distributions,” Extremes, vol. 6, no. 2, pp. 91–109, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. R. Ihaka and R. Gentleman, “R: a language for data analysis and graphics,” Journal of Computational and Graphical Statistics, vol. 5, no. 3, pp. 299–314, 1996.
  32. S. Cheng and L. Peng, “Confidence intervals for the tail index,” Bernoulli, vol. 7, no. 5, pp. 751–760, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet