Joint Distributions Associated with Compound Patterns in a Sequence of Markov Dependent Multistate Trials and Estimation Problems

Abstract

Let Λi ,1 ≤ i ≤ � be simple patterns, i.e., finite sequences of outcomes from a set Γ = {b1 ,b 2 ,...,b m} and let Λ be a compound pattern (a set ofdistinct simple patterns). In this paper, we studyjoint distributions of the waiting time until the r-th occurrence of the compound pattern Λ, and the numbers of each simple pat- tern observed at that time in the multistate Markov dependent trials. We provide methods for deriving the probabilitygenerating functions of the joint distributions under two types of counting schemes (non-overlap counting and overlap counting) for the compound pattern Λ. Besides, the present work is useful in elucidating the pri- marydifference between non-overlap counting and overlap counting. As applications, when Λ is a set of runs, the corresponding joint distributions are investigated and a practical example is mentioned. Also, the Chen-Stein approximation is derived for the waiting time distribution, and its asymptotic behaviour is discussed. Finally, we address the parameter estimation in the waiting time distributions of the compound pattern along with problems of identifiability.