Masahito Hayashi et Shun Watanabe, INFORMATION GEOMETRY APPROACH TO PARAMETER ESTIMATION IN MARKOV CHAINS, Annals of statistics , 44(4), 2016, pp. 1495-1535
We consider the parameter estimation of Markov chain when the unknown transition matrix belongs to an exponential family of transition matrices. Then we show that the sample mean of the generator of the exponential family is an asymptotically efficient estimator. Further, we also define a curved exponential family of transition matrices. Using a transition matrix version of the Pythagorean theorem, we give an asymptotically efficient estimator for a curved exponential family.