0
$\begingroup$

Let $X_1,X_2,...,X_n$ be iid observations from a normal distribution with mean $\mu$ and variance $\sigma^2$, $\sigma^2>0$ is known and $\mu$ is an unknown real number. Let $g(\mu)=2\mu$ be the parameter of interest. I know that $(\bar X,S^2)$ is a complete sufficient statistics for ($\mu$,$\sigma^2$) where $\bar X$ is the sample mean and $S^2$ is the sample variance. I am a little confused here. How do I find the UMVUE of $g(\mu)$?

$\endgroup$
  • $\begingroup$ It is unclear where exactly you are stuck. $\endgroup$ – StubbornAtom Apr 17 at 21:53
  • $\begingroup$ UMVUE of $\mu$ is $\bar X$ so UMVUE of $2\mu$ is $2\bar X.$ $\endgroup$ – BruceET Apr 18 at 0:12
0
$\begingroup$

Using Lehmann-Shceffé Theorem, if $T$ is a Sufficient and Complete statistics for $\mu$ and $T^{*}$ $=$ $h(T)$ is unbiased for $g(\mu)$, then $T^{*}$ is UMVUE for $g(\mu)$.

The theorem applies to your case since $T$ $=$ $\overline{X}$ is Sufficient and Complete for $\mu$, hence $T^{*}$ = $2$$\overline{X}$ is UMVUE for $g(\mu)$ = $2$$\mu$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.