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Hello I am trying find the mean and variance of the methods of moments estimator for the distribution of $$f(x;\theta) = \frac{1}{\theta} (\frac{1}{x})^{\frac{1}{\theta} + 1} $$ for x > 1, 0 < $\theta$ < 1.

Using the first moment I found that $\bar{x}$ = $\frac{1}{1 - \theta}$ which implies the parameter estimator is $\hat\theta = 1 - \frac{1}{\bar{x}}$. I'm trying to find the mean and variance for $\hat\theta$ so that I can determine if it is an unbiased and/or consistent estimator but am unable to do so because I don't know the actual distribution this estimator follows. Is there another way to determine if this estimator is unbiased and/or consistent without computing its mean and variance or knowing its actual distribution?

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By Jensen inequality with convex function $\varphi(x)=\frac1x$ ($x>1$) we have $$ \mathbb E[\varphi(\bar x)]=\mathbb E\left[\frac{1}{\bar x}\right] > \varphi(\mathbb E[\bar x])=\varphi(\mathbb E[X_1])=\frac{1}{\mathbb E[X_1]}=1-\theta. $$ The inequality is strict since $\bar x$ is not a degenerate random variable and the function $\varphi$ is non-linear on $(1,\infty)$.

Then $$\mathbb E[\hat \theta] = 1- \mathbb E\left[\frac{1}{\bar x}\right] < 1-(1-\theta) = \theta.$$

This inequality means that the estimate is biased.

To prove consistency, use Khintchine's Law of large numbers.

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