Find asymptotic efficiency of MLE to UMVUE

Let $$\{X_i\}_{i=1}^n$$ be a sequence of i.i.d random variables with common pdf: $$f(x;a,\theta) =\theta a^\theta x^{-(\theta+1)} \boldsymbol 1_{(a,\infty)}(x) \, \,\text{; where } \theta, a > 0$$ I would like to find the asymptotic relative efficiency of the MLE of a with respect to the UMVUE of a. I believe I have already found the correct MLE for $$a$$ but I'm not sure how to find the efficiency in this case. From the likelihood it seems like the first order statistic is the MLE. $$\mathcal{L}(X,a,\theta) = \theta^n a^{n\theta} \prod_{i=1}^n\left( x_i^{-(\theta+1)} \boldsymbol \cdot 1_{(a,\infty)}(x_i)\right) \implies \min_{1 \leq i \leq n}(x_i) = \hat{a}$$

Typically I would use the score to find the Fisher Information and then take the ratio but I don't think that works here. Thanks for your help.

MLE of $$a$$ is indeed the first order statistic $$X_{(1)}=\min\limits_{1\le i\le n}X_i$$ because the likelihood is non-decreasing in $$a$$ subject to the restriction $$a. Because the population distribution is Pareto, you can verify that $$X_{(1)}$$ also has a Pareto distribution from which you can get its exact variance.

UMVUE of $$a$$ however depends on whether $$\theta$$ is known or not. In any case, it is found using the Lehmann-Scheffé theorem.

• If $$\theta$$ is known, then $$X_{(1)}$$ is a complete sufficient statistic and UMVUE of $$a$$ is of the form $$c(\theta) X_{(1)}$$ for some function $$c$$.

• If $$\theta$$ is not known, then $$\left(\prod\limits_{i=1}^n X_i,X_{(1)}\right)$$ or equivalently $$\left(U, X_{(1)}\right)$$ is a complete sufficient statistic where $$U=\sum\limits_{i=1}^n (\ln X_i-\ln X_{(1)})$$. Here $$U$$ has a certain Gamma distribution, and $$U$$ and $$X_{(1)}$$ can be shown to be independent. The resulting UMVUE of $$a$$ would be of the form $$g(U)X_{(1)}$$ for some function $$g$$.

Using the points above, you can find the exact variance of both UMVUE and MLE of $$a$$. Asymptotic relative efficiency of MLE with respect to UMVUE is then the limit of the ratio $$\operatorname{Var}(\hat a)/\operatorname{Var}(X_{(1)})$$ as $$n\to \infty$$ where $$\hat a$$ is UMVUE of $$a$$. Note that Fisher information is not usually defined for non-regular distributions like this where support of the distribution depends on the parameter of interest.

Since the MLE is the first order statistic it has the pdf: $$n \theta a^{n\theta} x^{-(n\theta+1)} \boldsymbol 1_{(a,\infty)}(x)$$.

Naturally then $$\displaystyle \mathbb{E}[X_{(1)}] = \frac{a n\theta}{n\theta-1 }$$, so that $$\displaystyle \frac{(n\theta-1)X_{(1)} }{n\theta } = T(x)$$ is the UMVUE by Lehmann-Scheffé.

$$\implies \frac{Var(T(x)) }{Var(X_{1})} = \left(\frac{(n\theta -1) }{n\theta }\right)^2 = 1 - \frac{2}{n \theta} + \frac{1}{n^2\theta^2}$$

So that the asymptotic efficiency is indeed 1.