Suppose that we have a time-homogeneous discrete-time Markov chain $(X_n)$. We want to estimate the transition probabilities $p_{ij} = \mathbb{P}[X_{n+1} = j \mid X_n = i]$.

In the case when we have full information, e.g. we observe $(X_1, X_2, \ldots, X_N)$, the maximum likelihood estimator is $$ \hat{p}_{ij} = \frac{ n_{ij} }{ \sum_{k=1}^{m} n_{ik} }, $$ where $n_{ik}$ is the number of times that the process moved from state $i$ to $k$. (I took the expression above from https://stats.stackexchange.com/a/14361/22409)

Unfortunately, in my case we are not able to observe all the transitions. For example, we may observe $X_1, X_6, X_{11}, X_{13}, X_{14}, X_{15}, \ldots$, that is some of the states, but the states that we observe are not regularly spaced.

I have two related questions:

  1. Mathematically, how do we estimate the transition probabilities given such data observations?
  2. What statistical software is available for estimating the transition probabilities?

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.