I'd like to ask for help on solving the following problem - I believe I solved the first part but I'm not sure how to proceed with the rest:
Let $X$ be an observation from a normal distribution with mean $\mu$ and variance 1. Find the UMVUE of $μ^2$ and the UMVUE of $P[|X| > 1]$. Compare these with the MLE’s of these parameters.
I started by identifying that $f_X(x)$ is part of the exponential family and thus X is a complete and sufficient statistic for $\mu$. Further, I found that $E[X^2] = 1+ \mu^2$ and so $X^2-1$ is an unbiased estimator for $\mu^2$ and a function of our complete and sufficient statistic. Therefore, $X^2-1$ is the UMVUE of $\mu^2$.
I am not sure how to find the UMVUE for $P[|X| > 1]$ and compare them with the MLEs. Thank you for any help.
Let's set the following estimator T for $\mathbb{P}[|X|>1]$
$$T=\mathbb{1}_{|X|>1}= \begin{cases} 0, & \text{if $\mathbb{P}[|X|\leq1]$} \\ 1, & \text{if $\mathbb{P}[|X|>1]$ } \end{cases}$$
Obviously $\mathbb{E}[T]=\mathbb{P}[|X|>1]$
Then T is unbiased and function of a CSS.
In order to get the MLE you can use invariance property