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Hello I am trying to see if the Maximum likelihood estimator I have found for the distribution of $$f(x;\theta) = \frac{1}{\theta} (\frac{1}{x})^{\frac{1}{\theta} + 1} $$ for x > 1, 0 < $\theta$ < 1.

is an unbiased and/or a consistent estimator of $\theta$. I have found the maximum likelihood estimator of this function to be $\hat{\theta} = \bar{y} $ for $ y = ln x_i$ The problem I seem to be having checking if this estimator is unbiased and/or consistent is finding the first and second moments of the estimator so that I can solve for the mean and variance of this estimator. How would I approach this type of problem, would I need to take a Taylor series estimate or is there some other way to find the expected value and variance of the mean of the natural logarithm of the distribution?

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First, let's find the distribution of $\ln x_i$. The CDF of $x_i$ is $$ F_{x_i}(x)=P\{x_i\le x\}=\int_1^x\frac1\theta\left(\frac1z\right)^{1/\theta+1}dz=1-\left(\frac1x\right)^{1/\theta}, \text{for $x\ge1$}. $$ So the CDF of $\ln x_i$ is $$ F_{\ln x_i}(x)=P\{\ln x_i\le x\}=P\{x_i\le e^x\}=1-e^{-x/\theta}, \text{for $\ln x_i\ge0$}. $$ This means that $\ln x_i$ is an exponential random variable with expected value $\theta$. Hence, the mean $\overline{\ln x}$ is an unbiased estimator of $\theta$.

Then we can apply the law of large numbers and conclude that $\overline{\ln x}$ converges in probability to its mean $\theta$, and therefore it is a consistent estimator of $\theta$.

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