No abstract

Approximations for an unknown density g in terms of a reference density fν and its associated orthonormal polynomials are discussed. The main application is the approximation of the density f of a sum S of lognormals which may have different variances or be dependent. In this setting, g may be f itself or a transformed density, in particular that of log S or an exponentially tilted density. Choices of reference densities fν that are considered include normal, gamma and lognormal densities. For the lognormal case, the orthonormal polynomials are found in closed form and it is shown that they are not dense in L2(fν ), a result that is closely related to the lognormal distribution not being determined by its moments and provides a warning to the most obvious choice of taking fν as lognormal. Numerical examples are presented and comparison are made to established approaches such as the Fenton-Wilkinson method and skew-normal approximations. Also extension to density estimation for statistical data sets and non-Gaussian copulas are outlined.

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance. The advantage is that PH distributions form a dense class and that problems having explicit solutions for exponential distributions typically become computationally tractable under PH assumptions. In the first part of this paper, fitting of PH distributions to human lifetimes is considered. The class of generalized Coxian distributions is given special attention. In part, some new software is developed. In the second part, pricing of life insurance products such as guaranteed minimum death benefit and high-water benefit is treated for the case where the lifetime distribution is approximated by a PH distribution and the underlying asset price process is described by a jump diffusion with PH jumps. The expressions are typically explicit in terms of matrix-exponentials involving two matrices closely related to the Wiener-Hopf factorization, for which recently, a Lévy process version has been developed for a PH horizon. The computational power of the method of the approach is illustrated via a number of numerical examples.

We study an unbiased estimator for the density of a sum of random variables that are simulated from a computer model. A numerical study on examples with copula dependence is conducted where the proposed estimator performs favourably in terms of variance compared to other unbiased estimators. We provide applications and extensions to the estimation of marginal densities in Bayesian statistics and to the estimation of the density of sums of random variables under Gaussian copula dependence.

TAIL ASYMPTOTICS OF LIGHT-TAILED WEIBULL-LIKE SUMSWe consider sums of n i.i.d. random variables with tailsclose to e xp{−x} for some > 1. Asymptotics developed by Rootzén 1987 and Balkema, Klüppelberg, and Resnick 1993 are discussed from the point of view of tails rather than of densities, using a some what differentangle, and supplemented with bounds, results on a random number N of terms, and simulation algorithms.