# Test Markov Chain properties of time series

First of all, please excuse if I don't use the proper terminology for this problem.

I have a markov chain composed by two states:

• When in state 1 the output is drawn from an exponential distribution with mean parameter mu_exp
• When in state 2 the output is drawn from a gaussian distribution with mean mu_gaus and standard deviation gamma_gaus

I have a time series which I assume as being ruled by that markov model.

My objective is to generate many random outcomes from that markov model and test its likelihood with the original time series.

Is there a test, like Kolmogorov-Smirnov or mann-whitney-wilcoxon to reject the null-hypotesis "The two series are outcomes of the same markov chain"?

Again, sorry for the terminology, I hope I was clear enough

• That looks difficult to me. What you describe is a "Hidden Markov Model". I don't know if there any nice formulas for the likelihood of such models. – StijnDeVuyst Mar 22 '15 at 21:13