Doctor of Philosophy
Statistics and Actuarial Sciences
Provost, Serge B.
Several advances are proposed in connection with the approximation and estimation of heavy-tailed distributions, some of which also apply to other types of distributions. It is first explained that on initially applying the Esscher transform to heavy-tailed density functions such as the Pareto, Student-t and Cauchy densities, one can utilize a moment-based technique whereby the tilted density functions are expressed as the product of a base density function and a polynomial adjustment. Alternatively, density approximants can be secured by appropriately truncating the distributions or mapping them onto compact supports. The validity of these approaches is corroborated by simulation studies. Extensions to the context of density estimation, in which case sample moments are employed in lieu of exact moments are discussed, and illustrative applications involving actuarial data sets are presented. Novel approaches involving making use of the Box-Cox transform in conjunction with empirical saddlepoint density estimates and generalized beta density functions are introduced for determining the endpoints of empirical distribution. Additionally, an iterative algorithm and a technique relying on approximating a function by means of Bernstein polynomials are proposed for obtaining smooth bona fide density functions. Finally, a polynomial adjustment is applied to a bivariate empirical saddlepoint estimate which is obtained from a sample estimate of the bivariate cumulant generating function. A significant contribution of this dissertation resides in the implementation of the proposed methodologies such as the constrained estimation of the four parameters of the generalized beta distribution and the adjusted bivariate empirical saddlepoint density estimation technique in the symbolic computing package Mathematica.
Kang, Sang Jin, "Advances in the Modeling of Heavy-tailed Distributions" (2018). Electronic Thesis and Dissertation Repository. 6003.
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