No abstract

Let ε be a (single or composite) pattern defined over a sequence of Bernoulli trials. This article presents a unified approach for the study of the joint distribution of the number S n of successes (and F n of failures) and the number X n of occurrences of ε in a fixed number of trials as well as the joint distribution of the waiting time T r till the rth occurrence of the pattern and the number S T r of successes (and F T r of failures) observed at that time. General formulae are developed for the joint probability mass functions and generating functions of (X n ,S n ), (T r ,S T r ) (and (X n ,S n ,F n ),(T r ,S T r ,F T r )) when X n belongs to the family of Markov chain imbeddable variables of binomial type. Specializing to certain success runs, scans and pattern problems several well-known results are delivered as special cases of the general theory along with some new results that have not appeared in the statistical literature before.

In this paper we consider an extension to the renewal or Sparre Andersen risk process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie-GumbelMorgenstern copula proposed by Journal, 2010, 3, 221-245]. We consider that the inter-arrival times follow the Erlang(n) distribution. By studying the roots of the generalized Lundberg equation, the Laplace transform of the expected discounted penalty function is derived and a detailed analysis of the Gerber -Shiu function is given when the initial surplus is zero. It is proved that this function satisfies a defective renewal equation and its solution is given through the compound geometric tail representation of the Laplace transform of the time to ruin. Explicit expressions for the discounted joint and marginal distribution functions of the surplus prior to the time of ruin and the deficit at the time of ruin are derived. Finally, for exponential claim sizes explicit expressions and numerical examples for the ruin probability and the Laplace transform of the time to ruin are given.

No abstract

No abstract