Estimating of the Renewal Function with Heavy-tailed Claims

Authors: Rassoul Abdelaziz

Abstract:

We develop a new estimator of the renewal function for heavy-tailed claims amounts. Our approach is based on the peak over threshold method for estimating the tail of the distribution with a generalized Pareto distribution. The asymptotic normality of an appropriately centered and normalized estimator is established, and its performance illustrated in a simulation study.

Keywords: Renewal function, peak-over-threshold, POT method, extremes value, generalized pareto distribution, heavy-tailed distribution.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1077283

References:

[1] Anderson, k., Athreya, K.B. (1987). A renewal theorem in the infinite mean case. Ann. Prob. 15 388-393. MR 88h:60154.
[2] Bebbington, M., Davydov, Y.and Zitikis, R. (2007) Estimating The Renewal Function when the Second Moment is Infinite, Stochastic Models, 23:27-48.
[3] Bingham, N., Goldie, C., and Teugels, J. (1987), Regular Variation, no. 27 in Encyclopedia of Mathematics and its Applications, Cambridge University Press.
[4] Embrechts, P., Kl¨uppelberg, C., and Mikosch, T. (1997), Modelling Extremal Events, Springer.
[5] Feller, W. (1971). An introduction to probability theory and its applications 2, 2nd ed. Wiley, New York. MR 42:5292
[6] Frees, Edward W. (1986). Nonparametric renewal function estimation. Ann. Statist. 14, 1366-1378.
[7] Gr¨ubel, R., Pitts, S.M. Nonparametric estimation in renewal theory: the empirical renewal function. Ann. Statist. 1993, 21, 1431-1451.
[8] Johansson, J. (2003) Estimating the mean of heavy-tailed distributions, Extremes 6, 91-131.
[9] Lehmann, E. L. and Casella, G., 1998. Theory of Point Estimation. Springer.
[10] Levy, J. B. and Taqqu, M. S. (2000), "Renewal Reward Processes with Heavy-Tailed inter-Renewal Times and Heavy-Tailed Rewards," Bernoulli, 6, 23-44.
[11] Markovitch, N.M., Krieger, U.R. (2000). Nonparametric estimation of long-tailed density functions and its application to the analysis of World Wide Web traffic. Performance Evaluation 42, 205-222.
[12] Mohan, N. R. (1976). Teugels renewal theorem and stable laws, Ann. Prob. 4 863-868. MR 54:6312.
[13] Pipiras, V. and Taqqu, M. S. (2000), "The Limit of a Renewal Reward Process with Heavy-Tailed Rewards is not a Linear Fractional Stable Motion," Bernoulli, 6, 607-614.
[14] Resnick, S. (1987), Extreme Values, Regular Variation and Point Processes, Springer-Verlag.
[15] Schneider, H., Lin, B.S., O-Cinneide, C. Comparison of nonparametric estimators for the renewal function. J. Royal Statist. Soc., Series C. 1990, 39, 55-61.
[16] Seneta, E. (1976), Regularly Varying Functions, no. 508 in Lecture Notes in Mathematics, Springer-Verlag.
[17] Sgibnev, M.S. Renewal theorem in the case of an infinite variance. Siberian Math. J. 1981, 22, 787-796.
[18] Smith, R. (1987), "Estimating Tails of Probability Distributions," The Annals of Statistics, 15, 1174-1207.American Petroleum Institute, Technical Data Book - Petroleum Refining, 5th edition, 1992.