Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T02:48:56.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Second Order Bayes Prediction of Functionals of Exponential Dispersion Distributions and an Application to the Prediction of the Tails

Published online by Cambridge University Press:  17 April 2015

Zinoviy Landsman
Zinoviy Landsman*
Affiliation:
Department of Statistics, University of Haifa, Mount Carmel, Haifa 31905, Israel. E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Second order Bayes estimators, being the main tool in second order optimal statistical theory, provide a natural basis for a new approach to the problem of the prediction of functions of expectation functional for members of an exponential dispersion family. A general formula, providing such prediction up to the term of the order 1/n, is suggested and the application to the problem of the prediction of the tail of distributions is demonstrated. The results are illustrated with normal and gamma claim sizes. The numerical experiment demonstrates the high effectiveness of the approach even for small sample sizes.

Keywords

Second order Bayes estimatorpredictive tailpredictive meancredibility formulalower truncated meanvariance functions
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 34 , Issue 2 , November 2004 , pp. 285 - 298
Copyright
Copyright © ASTIN Bulletin 2004

References

Aitchinson, J., Dunsmore, I.R. (1975) Statistical Prediction Analysis . Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. (1989) Asymptotic Techniques for Use in Statistics. Chapman and Hall, London, New York CrossRefGoogle Scholar
Bickel, P.J. (1981) Minimax estimation of the mean of a normal distribution when the parameter space is restricted. Ann. Statist., 9(6), 1301–1309.CrossRefGoogle Scholar
Brown, L.D. (1986) Fundamentals of Statistical Exponential Families. Lecture Notes – Monograph series, Hayward, CA.Google Scholar
Bühlmann, H. (1967) Experience rating and probability. The Astin Bull. 4, 199–207.CrossRefGoogle Scholar
Diaconis, P. and Ylvisaker, D. (1979) Conjugate priors for exponential families. The Annals of Statist. 7, 269–281.CrossRefGoogle Scholar
Fuller, W.A. (1996). Introduction to Statistical Time Series, Wiley. New York. Google Scholar
Jewell, W.S. (1974) Credible means are exact Bayesian for exponential families. Astin Bull. 8, 77–90.CrossRefGoogle Scholar
Jorgensen, B. (1986) Some properties of exponential dispersion models. Scan. J. Statist. 13, 187–198.Google Scholar
Jorgensen, B. (1987) Exponential dispersion models (with discussion). J. Roy. Statist. Soc. Ser. B 49, 127–162.Google Scholar
Jorgensen, B. (1992) Exponential dispersion models and extensions: A review. Internat. Statist. Rev. 60, 5–20.CrossRefGoogle Scholar
Jorgensen, B. (1997) The Theory of Dispersion Models. Chapman and Hall, London. Google Scholar
Kaas, R., Dannenburg, D. and Goovaerts, M. (1997) Exact credibility for weighted observations. Astin Bull . 27(2), 287–295.CrossRefGoogle Scholar
Landsman, Z. (1996) Sample quantiles and additive statistics: information, sufficiency, estimation. Journal of Statistical Planning and Inference, 52, 93–108.CrossRefGoogle Scholar
Landsman, Z. (2001) Second order minimax estimation of the mean value for Exponential Dispersion models. Journal of Statistical Planning and Inference, 98, 57–71.10.1016/S0378-3758(00)00315-3CrossRefGoogle Scholar
Landsman, Z. (2002) Credibility theory: A new view from the theory of second order optimal statistics. Insurance: Mathematics & Economics, 30, 351–362.Google Scholar
Landsman, Z. and LEVIT, B. (1990) The second order minimax estimation: nuisance parameters and hypoelliptic operators. Probability Theory and Mathematical Statistics, Proceedings of the Fifth International Vilnius Conference, 2, 47–58, VSP, Utrecht. Google Scholar
Landsman, Z. and Levit, B. (1991) Second order asymptotic minimax estimation in the presence of a nuisance parameter. Problems of Information Transmissions, 26(3), 226–244.Google Scholar
Landsman, Z. and Makov, U. (1998) Exponential dispersion models and credibility. Scand. Actuarial J., 1, 89–96.CrossRefGoogle Scholar
Landsman, Z. and Makov, U. (1999) On stochastic approximation and credibility. Scand. Actuaial J., 1, 15–31.CrossRefGoogle Scholar
Levit, B. (1980) Second order minimaxity. Theory of Prob. Appl. 25, 3, 561–576.Google Scholar
Levit, B. (1982) Second order minimax estimation and positive solutions of elliptic equations. Theory Prob. Appl. 27(3), 525–546.Google Scholar
Lindley, D.V. (1980) Approximate Bayesian Methods. Bayesian Statistics. Proceedings of the 1st International Meetings held in Valencia, 223–237.Google Scholar
Nelder, J.A. and Verrall, R.J. (1997) Credibility theory and generalized linear models. Astin Bull., 27(3), 71–82.CrossRefGoogle Scholar
Nelder, J.A. and Wedderburn, R.W.M. (1972) Generalized linear models. J. Roy. Statist. Soc. Ser. A 135, 370–384.CrossRefGoogle Scholar
Tierney, L. and Kadane, J.B. (1986) Accurate approximations for posterior moments and marginal densities JASA, 81, 393, 82–86.CrossRefGoogle Scholar
Tweedie, M.C.K. (1984) An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions. Proceedings of the Indian Statistical Golden Jubilee International Conference (Eds. Ghosh, J.K. and Roy, J.), 579–604. Indian Statistical Institute.Google Scholar
Young, V.R. (1998) Credibility using a loss function from spline theory: Parametric models with a one-dimensional sufficient statistic. North American Actuarial Journal, 2, 1, 101–117.CrossRefGoogle Scholar