A compound negative binomial distribution with mutative termination conditions based on a change point

In this paper, an extended negative binomial distribution called NBM is proposed by considering mutative termination conditions based on a change point. If the condition A is satisfied before the occurrence of the change point, the trial is terminated according to condition A. Otherwise, if the condition A does not happen and the change point is satisfied, the termination condition of the trial changes from condition A to condition B. We consider the conditions under which the resulting distribution can be degenerated to the existing negative binomial distributions and other new negative binomial distributions. The finite Markov chain imbedding approach is employed to derive the new negative binomial distribution (NBM) and to obtain the related probabilistic indexes. Furthermore, we study the distribution of the compound negative binomial distribution with mutative termination conditions based on a change point (CNBM) by means of phase-type representations.