Copula models are becoming increasingly popular for modelling dependencies between random variables. The range of their recent applications includes such fields as analysis of extremes in financial assets and returns; failure of paired organs in health science; reliability studies; and human mortality in insurance.This paper gives a brief overview of the principles of construction of such copula models as Gaussian, Student, and Archimedean. The latter includes Frank, Clayton, and stable (Gumbel-Hougaard) families. The emphasis is on application of copula models to joint last survivor analysis.The main example discussed in this paper deals with the mortality of spouses, known to be associated through such factors as common disaster, common lifestyle, or the broken-heart syndrome. These factors suggest modelling dependence of spouses' lives on both calendar date scale and age-at-death scale. This dependence structure suggests a different treatment than that for problems of survival analysis such as paired organ failure or twins' mortality.Construction of a conditional Bayesian copula model is further generalized in view of the relationship between the joint first life and last surviror probabilities. A numerical example is considered, involving the implementation of Markov chain Monte Carlo algorithms using WinBUGs. Maximum likelihood estimation for this model using (5) is carried out in Reference [3], whereFrank's copula and Gompertz marginals are also considered. Bayesian estimation with exponential and normal priors for Weibull distribution parameters, beta prior for the association, and the copula functions of three forms: stable, Frank's and Clayton's, is performed in Reference [18].This approach is effective. However, it sometimes proves to be insufficient depending on the type of association between the paired lives. Apparently, the source of this insufficiency is the problem of dimensionality. We use a bivariate survival function Sðx 1 ; x 2 Þ to model the behaviour of trivariate joint first life and last survivor functions. Why are these functions trivariate?

This book is a self-contained introduction to the theory of distributions, sometimes called generalized functions. Most books on this subject are either intuitive or else rigorous but technically demanding. Here, by concentrating on the essential results, the authors have introduced the subject in a way that will most appeal to non-specialists, yet is still mathematically correct. Topics covered include: the Dirac delta function, generalized functions, dipoles, quadrupoles, pseudofunctions and Fourier transforms. The self-contained treatment does not require any knowledge of functional analysis or topological vector spaces; even measure theory is not needed for most of the book. The book, which can be used either to accompany a course or for self-study, is liberally supplied with exercises. It will be a valuable introduction to the theory of distributions and their applications for students or professionals in statistics, physics, engineering and economics.

We derive a differential equation and recursive formulas of Sheffer polynomial sequences utilizing matrix algebra. These formulas provide the defining characteristics of, and the means to compute, the Sheffer polynomial sequences. The tools we use are well-known Pascal functional and Wronskian matrices. The properties and the relationship between the two matrices simplify the complexity of the generating functions of Sheffer polynomial sequences. This work extends He and Ricci's work (2002) to a broader class of polynomial sequences, from Appell to Sheffer, using a different method. The work is self-contained.

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