## Mixtures of tails in clustered automobile collision claims.(English)Zbl 0864.62071

Summary: Knowledge of the tail shape of claim distributions provides important actuarial information. This paper dicusses how two techniques commonly used in assessing the most appropriate underlying distribution can be usefully combined. The maximum likelihood approach is theoretically appealing since it is preferable to many other estimators in the sense of best asymptotic normality. Likelihood based tests are, however, not always capable to discriminate among non-nested classes of distributions. Extremal value theory offers an attractive tool to overcome this problem. It shows that a much larger set of distributions is nested in their tails by the so-called tail parameter.
This paper shows that both estimation strategies can be usefully combined when the data generating process is characterized by strong clustering in time and size. We find that the extreme value theory is a useful starting point in detecting the appropriate distribution class. Once that has been achieved, the likelihood-based EM-algorithm is proposed to capture the clustering phenomena. Clustering is particularly pervasive in actuarial data. An empirical application to a four-year data set of Dutch automobile collision claims is therefore used to illustrate the approach.

### MSC:

 62P05 Applications of statistics to actuarial sciences and financial mathematics 60G70 Extreme value theory; extremal stochastic processes
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### References:

 [1] Aiuppa, T.A., Evaluation of Pearson curves as an approximation of the maximum probable annual aggregate loss, Journal of risk and insurance, 55, 425-439, (1988) [2] Barton, D.E.; David, F.N.; Merrington, M., Table for the solution of the exponential equation exp(b)−b/(1 − p) = 1, Biometrika, 50, 169-176, (1963) · Zbl 0117.14903 [3] Beirlant, J.; Teugels, J.L., Modeling large claims in non-life insurance, Insurance: mathematics and economics, 11, 17-30, (1992) · Zbl 0753.62074 [4] Beirlant, J.; Teugels, J.L.; Vynckier, P., Extremes in non-life insurance, () [5] Benktander, G.; Segerdahl, C.O., On the analytical representation of claim distributions with special reference to excess of loss reinsurance, Comptes rendus du XVI congres international d’actuaires, I, 626-648, (1960) [6] Boos, D.D., Using extreme value theory to estimate large percentiles, Technometrics, 26, 33-39, (1984) [7] Cummins, J.D.; Freifelder, L.R., A comparative analysis of alternative maximum probable yearly aggregate loss estimators, Journal of risk and insurance, 45, 27-52, (1978) [8] Cummins, J.D.; Dionne, G.; McDonald, J.B.; Pritchett, B.M., Applications of the GB2 family of distributions in modelling insurance loss processes, Insurance: mathematics and economics, 9, 257-272, (1990) [9] Dekkers, A.L.M.; Einmahl, J.H.J.; de Haan, L., A moment estimator for the index and large quantile estimation, The annals of statistics, 3, 743-756, (1989) · Zbl 0701.62029 [10] Dempster, A.E.; Laird, N.; Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm, Journal of the royal statistical society, series B, 39, 1-38, (1977) · Zbl 0364.62022 [11] Galambos, J., The selection of the domain of attraction of an extreme value distribution from a set of data, () · Zbl 0672.62035 [12] Galambos, J.; Lechner, J.; Simiu, E., Extreme value theory and applications, (1994), Kluwer Academic Publishers Dordrecht [13] Hall, P., Asymptotic properties of the bootstrap for heavy-tailed distributions, Annals of probability, 18, 1342-1360, (1990) · Zbl 0714.62035 [14] Hasover, A.M.; Wang, Z., A test for extreme value domain of attraction, Journal of the American statistical association, 87, 171-177, (1992) [15] Hill, B.M., A simple general approach to inference about the tail of a distribution, Annals of statistics, 3, 1163-1173, (1975) · Zbl 0323.62033 [16] Hogg, R.V.; Klugman, S.A., On the estimation of long tailed skewed distributions with actuarial applications, Journal of econometrics, 23, 92-102, (1983) [17] Koedijk, K.G.; Schafgans, M.A.; de Vries, C.G., The tail index of exchange rate returns, Journal of international economics, 29, 93-108, (1990) [18] Loretan, M.; Phillips, P.C.B., Testing the covariance stationarity of heavy-tailed time series, Journal of empirical finance, 1, 211-248, (1994) [19] Leadbetter, M.R.; Lindgren, G.; Rootzén, H., Extremes and related properties of random sequences and processes, (1983), Springer Verlag New York · Zbl 0518.60021 [20] Pentikäinen, T., Approximate evaluation of the distribution function of aggregate claims, Astin bulletin, 17, 15-39, (1987) [21] Ruud, P.A., Extensions of estimation methods using the EM algorithm, Journal of econometrics, 49, 305-341, (1991) · Zbl 0742.62106 [22] Tanner, M., Tools for statistical inference: methods for the exploration of posterior distribution and likelihood functions, (1993), Springer-Verlag New York · Zbl 0777.62003 [23] Weba, M., Fitting a parametric distribution for large claims in case of censored or partitioned data, Insurance: mathematics and economics, 12, 155-165, (1993) · Zbl 0779.62098 [24] White, H., Regularity conditions for Cox’s test of non-nested hypothesis, Journal of econometrics, 301-318, (1982) · Zbl 0539.62032 [25] Willmot, G.E., The total claims distribution under inflationary conditions, Scandinavian actuarial journal, 1-12, (1989) · Zbl 0679.62094 [26] Zelterman, D., A semiparametric bootstrap technique for simulating extreme order statistics, Journal of the American statistical association, 88, 477-485, (1993) · Zbl 0774.62044
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