8.2.4 Asymptotic Properties of MLEs
Asymptotic Properties of MLEs
Let $X_1$, $X_2$, $X_3$, $...$, $X_n$ be a random sample from a distribution with a parameter $\theta$. Let $\hat{\Theta}_{ML}$ denote the maximum likelihood estimator (MLE) of $\theta$. Then, under some mild regularity conditions,- $\hat{\Theta}_{ML}$ is asymptotically consistent, i.e., \begin{align} \lim_{n \rightarrow \infty} P(|\hat{\Theta}_{ML}-\theta|>\epsilon)=0. \end{align}
- $\hat{\Theta}_{ML}$ is asymptotically unbiased, i.e., \begin{align} \lim_{n \rightarrow \infty} E[\hat{\Theta}_{ML}]=\theta. \end{align}
- As $n$ becomes large, $\hat{\Theta}_{ML}$ is approximately a normal random variable. More precisely, the random variable \begin{align} \frac{\hat{\Theta}_{ML}-\theta}{\sqrt{\mathrm{Var}(\hat{\Theta}_{ML})}} \end{align} converges in distribution to $N(0,1)$.