Abstract
Several advances are made in connection with the approximation and estimation of heavy-tailed distributions. It is first explained that on initially applying the Esscher transform to heavy-tailed density functions such as the Pareto, Studentt and Cauchy, said densities can be approximated by employing a certain moment-based methodology. Alternatively, density approximants can be obtained by appropriately truncating such distributions or mapping them onto finite supports. These techniques are then extended to the context of density estimation, their validity being demonstrated by means of simulation studies. As well, illustrative actuarial examples are presented.
Highlights
Various density approximation techniques that are associated with the moments or the cumulants of a distribution have been proposed in the statistical literature
Johnson curves, which are described in Elderton and Johnson (1969), make use of the first four moments of a distribution to approximate a density on the basis of a system of frequency curves whose support can be finite, infinite or semi-infinite
This section explains how the density approximation methodology that is based on exponential tilting, which is discussed in Section 3, is applied when a sample of observations is available
Summary
Various density approximation techniques that are associated with the moments or the cumulants of a distribution have been proposed in the statistical literature. Jensen (1995) extensively covered the application of this method to various types of random variables, including i.i.d. sums, compound sums, Markov chains, and sums of independent but not necessarily identically distributed variables. In this instance, the density approximant is expressed as f (x) = √ 1 exp [K(s) − sx]. The main advantage of the saddlepoint approximation is its accuracy in the tails of the target density This approach may leave something to be desired in the case of multimodal distributions.
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