An Introduction to Probability and Statistical Inference, 2nd Edition

9.2 Some Properties of MLE's

Refer to Example 3 and suppose that we are interested in estimating the probability that 0 events occur; that is, $P_{\theta }\left(X=0\right)=\frac{{\text{e}}^{-\theta }{\theta }^0}{0!}={\text{e}}^{-\theta }$, call it $g_1\left(\mathit{\theta }\right)$ Thus, the estimated quantity is a function of θ rather than θ itself. Next, refer to Example 4 and recall that, if $X\sim f\left(x;\theta \right)=\theta {\text{e}}^{-\theta x},x>0$, then $E_{\theta }X=1/\theta$. Thus, in this case it would ...