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Robust and efficient estimation of the tail index of a single-parameter Pareto distribution. (English) Zbl 1083.62505

Summary: Estimation of the tail index parameter of a single-parameter Pareto model has wide application in actuarial and other sciences. Here we examine various estimators from the standpoint of two competing criteria: efficiency and robustness against upper outliers. With the maximum likelihood estimator (MLE) being efficient but nonrobust, we desire alternative estimators that retain a relatively high degree of efficiency while also being adequately robust. A new generalized median type estimator is introduced and compared with the MLE and several well-established estimators associated with the methods of moments, trimming, least squares, quantiles, and percentile matching. The method of moments and least squares estimators are found to be relatively deficient with respect to both criteria and should become disfavored, while the trimmed mean and generalized median estimators tend to dominate the other competitors. The generalized median type performs best overall. These findings provide a basis for revision and updating of prevailing viewpoints. Other topics discussed are applications to robust estimation of upper quantiles, tail probabilities, and actuarial quantities, such as stop-loss and excess-of-loss reinsurance premiums that arise concerning solvency of portfolios. Robust parametric methods are compared with empirical nonparametric methods, which are typically nonrobust.

MSC:

62F10 Point estimation
62F35 Robustness and adaptive procedures (parametric inference)
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
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References:

[1] Arnold B.C., Pareto Distributions (1983) · Zbl 1169.62307
[2] Beirlant J., Practical Analysis of Extreme Values (1996) · Zbl 0888.62003
[3] Brazauskas V., Robust Estimation of Tail Parameters for Two-Parameter Pareto and Exponential Models via Generalized Quantile Statistics (1999) · Zbl 0979.62016
[4] Choudhury J., Journal of Statistical Planning and Inference 19 pp 269– (1988) · Zbl 0662.62051
[5] Daykin C.D., Practical Risk Theory for Actuaries (1994) · Zbl 1140.62345
[6] Dekkers A.L.M., Annals of Statistics 17 pp 1795– (1989) · Zbl 0699.62028
[7] Donoho D.L., In A Festschrift for Erich pp 157– (1983)
[8] Gather U., Communications in Statistics, Part A–Theory and Methods 15 pp 2323– (1986) · Zbl 0603.62041
[9] Gerber H.U., An Introduction to Mathematical Risk Theory (1979) · Zbl 0431.62066
[10] Gomes C.P., In Principles and Practice of Constraint Programming CP-97 pp 121– (1997)
[11] Hampel F.R., Annals of Mathematical Statistics 42 pp 1887– (1971) · Zbl 0229.62041
[12] Harter H.L., Annals of Mathematical Statistics 32 pp 1078– (1961) · Zbl 0115.36403
[13] Hill B.M., Annals of Statistics 3 pp 1163– (1975) · Zbl 0323.62033
[14] Hodges J.L., Proceedings of Fifth Berkely Symposium on Mathematical Statistics and Probability 1 pp 163– (1967)
[15] Johnson N.L., Continuous Univariate Distributions, 2. ed. (1994) · Zbl 0811.62001
[16] Kimber A.C., Applied Statistics 32 pp 7– (1983)
[17] Kimber A.C., Journal of Statistical Computation and Simulation 18 pp 273– (1983) · Zbl 0527.62038
[18] Klugman S.A., Loss Models: From Data to Decisions (1998) · Zbl 0905.62104
[19] Koutrouvelis I.A., Communications in Statistics, Part A–Theory and Methods 10 pp 189– (1981) · Zbl 0461.62029
[20] Lehmann E.L., Theory of Point Estimation (1983) · Zbl 0522.62020
[21] Quandt R.E., Metrika 10 pp 55– (1966)
[22] Saleh A.K. Md. E., Annals of Mathematical Statistics 37 pp 143– (1966) · Zbl 0256.62034
[23] Sarhan A.E., Annals of Mathematical Statistics 34 pp 102– (1963) · Zbl 0109.37703
[24] Serfling R.J., Approximation Theorems of Mathematical Statistics (1980) · Zbl 0538.62002
[25] Serfling R., Annals of Statistics 12 pp 76– (1984) · Zbl 0538.62015
[26] Willemain T.R., Communications in Statistics, Part B?Simulation and Computation 21 pp 1043– (1992) · Zbl 04510679
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