# Markov transition probabilities and negative binomial distribution.

A realization of a Markov process generates a sequence of interval lengths between transition from one state to another. A natural way of modeling the distribution of the lengths is as a negative binomial distribution. Is there any relatively easy way to estimate the two parameters characterizing the negative binomial distribution from the n * (n - 1) parameters characterizing the Markov process? I am particularly interested in small values of n. I have data from which I can estimate these parameters, but I suspect that the process is not well modeled as a simple Markov process. I would appreciate references to any articles discussing this topic and suggesting different modeling approaches.

To clarify: I have data sets of time intervals from which I can easily obtain both the parameters of the negative binomial distribution fitting the data, and estimates of the Markov transition matrix. I can then use the Markov transition matrix to generate a set of time intervals which is used as the basis of a Monte Carlo estimation of the parameters of the negative binomial distribution corresponding to the Markov transition matrix. I'd like to be able to get those parameters without doing the Monte Carlo estimation. Subsequent investigation suggests that I should use fluctuation analysis methods to characterize the behavior of the data.

• Did you mean that the negative binomial distribution is the invariant distribution associated with your Markov process? – an12 Feb 23 '13 at 23:41
• I am under the impression he considers a continuous-time Markov chain with a negative binomial distribution between two consecutive jumps. That sounds strange. – Stéphane Laurent Feb 24 '13 at 22:09
• "A realization of a Markov process generates a sequence of interval lengths between transition from one state to another." These lengths are always exponentially distributed, hence to get other lengths distributions one must leave the realm of Markov processes. If this is what you have in mind, you might want to explain in more details. – Did May 10 '13 at 10:51