On least squares estimates of an exponential tail coefficient

Statistics and Risk Modeling

Volumn 14, Issue 4, 1996, Pages 353-372

  Schultze, J ^a   Steinebach, J ^a  

^a PHILIPPS UNIVERSITY MARBURG   (Germany)

Author keywords

adjustment coefficient in risk theory; consistency; Estimation of exponential tail coefficient; Hill estimate; least squares estimate

Indexed keywords

EID: 0242280328     PISSN: 21931402     EISSN: 21967040     Source Type: Journal    
DOI: 10.1524/strm.1996.14.4.353     Document Type: Article

References (24)

1. Bacro, J. N. & Brito, M. (1993). Strong limiting behaviour of a simple tail Pareto-index estimator. Statist. Decis. 3, 133-143.
2. B. ártfai, P. (1970). Limes superior S ötze f ür die Wartemodelle. Studia Sci. Math. Hungar. 5, 317-325.
3. Cohen, J.W. (1969). The Single Server Queue. North Holland, Amsterdam.
4. Csörgö, M. & Steinebach, J. (1991). On the estimation of the adjustment coefficient in risk theory via intermediate order statistics. Insur. Math. Econom. 10, 37-50.
5. Csörgö, S. & Teugels, J. L. (1990). Empirical Laplace transforms and approximation of compoud distributions. J. Appl. Prob. 27, 88-101.
6. Csörgö, S., Deheuvels, P. & Mason, D. (1985). Kernel estimates of the tail index of a distribution. Ann. Statist. 13, 1050-1077.
7. Csörgö, S., Haeusler, E. & Mason, D. M. (1991). The asymptotic distribution of extreme sums. Ann. Probab. 19, 783-811.
8. de Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tract 32, Math. Centre, Amsterdam.
9. de Haan, L. & Resnick, S. I. (1980). A simple asymptotic estimate for the index of a stable distribution. J. R. Statist. Soc. B 42 No. 1, 83-87.
10. Deheuvels, P., Haeusler, E. & Mason, D. M. (1988). Almost sure convergence of the Hill estimator. Math. Proc. Camb. Phil. Soc. 104, 371-381.
11. Deheuvels, P., Haeusler, E. & Mason, D. M. (1990). On the almost sure behavior of sums of extreme values from a distribution in the domain of attraction of a Gumbel law. Bull. Sc. Math. Ser. 2, 114, 61-95.
12. Deheuvels, P.& Steinebach, J. (1990). On some alternative estimates of the adjustment coefficient in risk theory. Scand. Actuarial J. 1990, 135-159 (1991).
13. Grandell, J. (1979). Empirical bounds for ruin probabilties. Stoch. Processes Appl. 8, 243-255.
14. Grandell, J. (1991). Aspects of Risk Theory. Springer-Verlag, Berlin, Heidelberg, New York.
15. Haeusler, E.& Teugels, J.L. (1985). On asymptotic normality of Hill's estimator for the exponent ot regular variation. Ann. Statist. 13, 743-756.
16. Herkenrath, U. (1986). On the estimation of the adjustment coefficient in risk theory by means of stochastic approximation procedures. Insur. Math. Econom. 5, 305-313.
17. Hill, B. M. (1975). A simple approach to inference about the tail of a distribution. Ann. Statist. 3, 1163-1174.
18. Lo, G. S. (1989). A note on the asymptotic normality of sums of extreme values. J. Statist. Plann. Infer. 22, 127-136.
19. Mason, D. M. (1982). Laws of large numbers for sums of extreme values. Ann. Probab. 10, 754-764.
20. Novak, S.Y.& Utev, S.A. (1990). Asymptotics of the distribution of the ratio of sums of random variables. Siberian Math. J. 31, 781-788 (1991).
21. Richter,W.-D., Steinebach, J.& Taube, S. (1993). On a class of estimators for an exponential tail coefficient with applications in risk theory. Statist. Decis. Suppl. Issue No. 3, 145-173.
22. Schultze, J. (1994). Eine Klasse von Schätzern für den Anpassungskoeffizienten in der Risikotheorie. Diplomarbeit, Fachbereich Mathematik, Philipps-Universiãtt Marburg.
23. Seneta, E. (1976). Regularly Varying Functions. Springer-Verlag, Berlin, Heidelberg, New York.
24. Viharos, L. (1993). Asymptotic distributions of linear combinations of extreme values. Acta Sci. Math. (Szeged) 58, 211-231.