Maximal Likelihood Estimator

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Definition

Introduction

Let’s say we observe a realization of $X_{0},X_{1},\ldots,X_{N}$ from a $\mbox{Markov}(\lambda,P)$ . How do we estimate the transition matrix $P$ ? Well lets define some things:

Likelihood

$L=P(X_{0}=x_{0},\ldots,X_{N}=x_{N})=\lambda_{x_{0}}p_{x_{0},x_{1}}\cdots p_{x_{N-1},x_{N}}$ ### Log-likelihood $\log(L)=\log(\lambda_{X_{0}})+\sum\limits_{i,j\in S}\left(\sum\limits_{k=0}^{N-1}\mathbb{1}_{\{X_{k}=i,X_{k+1}=j\}} \right)\log(p_{ij})$

For $i,j\in S$ the maximal likelihood estimator for $P$ is $\hat P_{ij}=\frac{\sum\limits_{k=0}^{N-1}\mathbb{1}_{\{X_{k}=i,X_{k+1}=j\}}}{\sum\limits_{k=0}^{N-1}\mathbb{1}_{\{X_{k}=i\}}}$

Assume $P$ is irreducible and positive recurrent. Then for any $i,j\in S$ $P\left(\lim_{n\to\infty}\hat P_{ij}=P_{ij}\right)=1$