@misc{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {Folded and log-folded-t distributions as models for insurance loss data}, doi = {http://www.tandfonline.com/doi/abs/10.1080/03461230903424199}, abstract = {A rich variety of probability distributions has been proposed in the actuarial literature for fitting of insurance loss data. Examples include: lognormal, log-t, various versions of Pareto, loglogistic, Weibull, gamma and its variants, and generalized beta of the second kind distributions, among others. In this paper, we supplement the literature by adding the log-folded-normal and log-folded-t families. Shapes of the density function and key distributional properties of the ‘folded’ distributions are presented along with three methods for the estimation of parameters: method of maximum likelihood; method of moments; and method of trimmed moments. Further, large and small-sample properties of these estimators are studied in detail. Finally, we fit the newly proposed distributions to data which represent the total damage done by 827 fires in Norway for the year 1988. The fitted models are then employed in a few quantitative risk management examples, where point and interval estimates for several value-at-risk measures are calculated.}, language = {en} } @misc{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {Authors 'Reply to' Letter to the Editor regarding folded models and the paper by Brazauskas and Kleefeld (2011)}, series = {Scandinavian Actuarial Journal}, journal = {Scandinavian Actuarial Journal}, number = {8}, issn = {0346-1238}, doi = {10.1080/03461238.2013.798070}, pages = {753 -- 757}, language = {en} } @misc{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {A statistical application of the quantile mechanics approach: MTM estimators for the parameters of t and gamma distribution}, series = {European Journal of Applied Mathematics}, volume = {23}, journal = {European Journal of Applied Mathematics}, number = {5}, issn = {0956-7925}, doi = {10.1017/S0956792512000137}, pages = {593 -- 610}, abstract = {In this paper, we revisit the quantile mechanics approach, which was introduced by Steinbrecher and Shaw (Steinbrecher, G. \& Shaw, W. T. (2008) Quantile mechanics. European. J. Appl. Math. 19, 87-112). Our objectives are (i) to derive the method of trimmed moments (MTM) estimators for the parameters of gamma and Student's t distributions, and (ii) to examine their large- and small-sample statistical properties. Since trimmed moments are defined through the quantile function of the distribution, quantile mechanics seems like a natural approach for achieving objective (i). To accomplish the second goal, we rely on the general large sample results for MTMs, which were established by Brazauskas et al. (Brazauskas, V., Jones, B. L. \& Zitikis, R. (2009) Robust fitting of claim severity distributions and the method of trimmed moments. J. Stat. Plan. Inference 139, 2028-2043), and then use Monte Carlo simulations to investigate small-sample behaviour of the newly derived estimators. We find that, unlike the maximum likelihood method, which usually yields fully efficient but non-robust estimators, the MTM estimators are robust and offer competitive trade-offs between robustness and efficiency. These properties are essential when one employs gamma or Student's t distributions in such outlier-prone areas as insurance and finance.}, language = {en} }