new user here self-studying some mathematical statistics. I came across this problem and am stuck.
Problem: Suppose that for $i = 1, ... , n$, the positive random variables $X_i$ are independent and each have the cumulative distribution function $G(x|\alpha) = 1 - e^{-\alpha x^2}$, for $x \geq 0, \alpha > 0.$
(i) Find the uniformly minimum-variance unbiased estimator (UMVUE) of $\sqrt{\alpha}$ based on $X_1, ... , X_n$, and explain why or why not the UMVUE obtained here is unique;
(ii) Solve for the Cramer-Rao Lower Bound (CRLB) for the variance of an unbiased estimator of $\sqrt{\alpha}$;
(iii) Find the variance of the UMVUE and determine if the variance of the UMVUE meets the CRLB, or if the CRLB can even be obtained at all.
Here is what I have tried and know:
Well, to start I know that the random variables $X_1, ..., X_n \stackrel{ind}{\sim} R$ with PDF $:= g(x|\alpha) = 2\alpha xe^{-\alpha x^2}$ have a liklihood function equal to $L(\textbf{x}| \alpha) = (2\alpha)^n(\Pi_{i = 1}^{n}x_i)e^{-\Sigma_{i = 1}^{n}\alpha x_i^2}$, which gives a log-liklihood function equal to $\mathscr{L}(x) = \ln(L(\textbf{x}|\alpha) = n\ln(2) + n\ln(\alpha) + \Sigma_{i=1}^{n}\ln(x_i) - \Sigma_{i=1}^{n}\alpha x_i^2$.
Taking the derivative of $\mathscr{L}(x)$ w.r.t $\alpha$ yields
$\mathscr{L}'(x) = \frac{d[\mathscr{L}(x)]}{d\alpha} = \frac{n}{\alpha} - \Sigma_{i=1}^{n}x_i^2$.
And while I'm aware this isn't provided to me, but from looking at this PDF and prior study I know this is that of a Rayleigh distribution (on the Wikipedia page, let $\sigma^2 = \frac{1}{2\alpha}$ and they're equivalent) which tells me it's expected value, but still, computing the expected value yields:
$E[R] = \int_0^\infty \! 2\alpha x^2e^{-\alpha x^2} \, \mathrm{d}x = \frac{\sqrt{\pi}} {2\sqrt{\alpha}}$. We use this value to define $\tau(\alpha)$. That is, let $\tau(\alpha) = \frac{\sqrt{\pi}} {2\sqrt{\alpha}}$
Now, I know from the textbook I'm using (Casella-Berger) that an estimator $W^{*}$ is a UMVUE of $\tau(\theta)$ if it satisfies $E_{\theta}[W^{*}] = \tau(\theta)$ for all $\theta$ and, for any other estimator $W$ with $E_{\theta}[W] = \tau(\theta)$, $Var_{\theta}(W^{*}) \leq Var_{\theta}(W)$.
Additionally, this distribution/PDF is a member of an exponential family, and has the statistic $T(\textbf{X}) = \Sigma_{i = 1}^{n}X_{i}^{2}$ which is a complete and sufficient statistic.
I know the Lehman-Scheffe theorem tells me that "Unbiased Estimators based on complete sufficient statistics are unique," and that I will have to use the Cramer-Rao inequality, but I'm just getting stuck on actually finding the UMVUE. Do I need to find the distribution of $T(\textbf{X})$? The next parts don't seem too bad once I have it, as it seems like I can maybe use Corollary 7.3.15 which deals with Attainment of the CRLB (on page 341 in chapter 7 if you have a copy), but for some reason I'm just getting stuck on actually finding the UMVUE and would be very grateful for some guidance. I feel like I have some of the pieces (or maybe not) and I'm just not seeing how to assemble them together and find the missing info I need. Thanks for taking the time to read this post and consider my question.