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?
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 $\theta$.