×

Bayesian estimators of the lognormal-Pareto composite distribution. (English) Zbl 1401.91120

Summary: In this paper, Bayesian methods with both Jeffreys and conjugate priors for estimating parameters of the lognormal-Pareto composite (LPC) distribution are considered. With Jeffreys prior, the posterior distributions for parameters of interest are derived and their properties are described. The conjugate priors are proposed and the conditional posterior distributions are provided. In addition, simulation studies are performed to obtain the upper percentage points of Kolmogorov-Smirnov and Anderson-Darling test statistics. Furthermore, these statistics are used to compare Bayesian and likelihood estimators. In order to clarify and advance the validity of Bayesian and likelihood estimators of the LPC distribution, well-known Danish fire insurance data-set is reanalyzed.

MSC:

91B30 Risk theory, insurance (MSC2010)
62E15 Exact distribution theory in statistics
62P05 Applications of statistics to actuarial sciences and financial mathematics
62F15 Bayesian inference
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beirlant, J., Teugels, J. L. & Vynckier, P. (1996). Practical analysis of extreme values. Leuven: Leuven University Press. · Zbl 0888.62003
[2] Beirlant, J., Joossens, E. & Segers, J. (2004). Generalized Pareto fit to the society of actuaries’ large claims database. North American Actuarial Journal8, 108-111. · Zbl 1085.62502
[3] Cooray, K. (2009). The Weibull-Pareto composite family with applications to the analysis of unimodal failure rate data. Communications in Statistics – Theory and Methods38, 1901-1915. · Zbl 1167.62021
[4] Cooray, K. & Ananda, M. A. (2005). Modeling actuarial data with a composite lognormal–Pareto model. Scandinavian Actuarial Journal5, 321-334. · Zbl 1143.91027
[5] D’Agostino, R. B. & Stephens, M. A. (1986). Goodness-of-fit techniques. New York: Marcel Dekker.
[6] Erdélyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. (1954). Tables of integral transforms. New York: McGraw-Hill.
[7] Gelman, A., Carlin, J. B., Sternand, H. S. & Rubin, D. B. (2003). Bayesian data analysis. 2nd ed.Boca Raton: Chapman and Hall.
[8] Hogg, R. V. & Klugman, S. A. (1984). Loss distributions. New York: John Wiley.
[9] Hossack, I. B., Pollard, J. H. & Zehnwirth, B. (1983). Introductory statistics with applications in general insurance. Cambridge: Cambridge University Press. · Zbl 0532.62080
[10] Jeffreys, H. (1961). Theory of probability. 3rd ed.London: Oxford University Press. · Zbl 0116.34904
[11] Klugman, S. A., Panjer, H. H. & Willmot, G. E. (1998). Loss models from data to decisions. New York: John Wiley. · Zbl 0905.62104
[12] Lehmann, E. L. & Casella, G. (1998). Theory of point estimation. New York: Springer. · Zbl 0916.62017
[13] McNeil, A. (1997). Estimating the tails of loss severity distributions using extreme value theory. ASTIN Bulletin27, 117-137.
[14] Patrik, G. (1980). Estimating casualty insurance loss amount distributions. Proceedings of the Casualty Actuarial Society LXVII, 57-109.
[15] Pigeon, M. & Denuit, M. (2011). Composite lognormal–Pareto model with random threshold. Scandinavian Actuarial Journal10, 177-192. · Zbl 1277.62258
[16] Preda, V. & Ciumara, R. (2006). On composite models: Weibull–Pareto and lognormal–Pareto. -A comparative study-. Romanian Journal of Economic Forecasting3, 32-46.
[17] Resnick, S. I. (1997). Discussion of the Danish data on large fire insurance losses. ASTIN Bulletin27, 139-151.
[18] Scollnik, D. P. M. (2007). On composite lognormal–Pareto models. Scandinavian Actuarial Journal7, 20-33. · Zbl 1146.91028
[19] Scollnik, D. P. M. & Sun, C. (2012). Modeling with Weibull–Pareto models. North American Actuarial Journal16, 260-272. · Zbl 1291.62186
[20] Teodorescu, S. (2010). On the truncated composite lognormal–Pareto model. Mathematical Reports (Bucureşti.)12, 71-84. · Zbl 1224.60022
[21] Teodorescu, S. & Panaitescu, E. (2009). On the truncated composite Weibull–Pareto model. Mathematical Reports (Bucureşti.)11, 259-273. · Zbl 1199.60041
[22] Teodorescu, S. & Vernic, R. (2009). Some composite exponential–Pareto models for actuarial prediction. Romanian Journal of Economic Forecasting12, 82-100.
[23] Zellner, A. (1971). Bayesian and non-Bayesian analysis of the log-normal distribution and log-normal regression. Journal of the American Statistical Association66, 327-330. · Zbl 0226.62064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.