The maximum likelihood estimator for a Pareto Distribution is given by:
$$\alpha = \frac{n}{\sum_\limits{i=1}^n ln x_i - ln x_{min}}$$
I want to find the expected value of $\alpha$ but don't know how. I tried to:
$$E[\alpha] = E[n(\sum_\limits{i=1}^n ln(\frac{x_i}{x_{min}})^{-1}] = nE[\frac{1}{\sum_\limits{i=1}^n ln(\frac{x_i}{x_{min}})}]$$
And then i got stuck. The only thing i can think of is to break somehow $\frac{1}{\sum_\limits{i=1}^n ln(\frac{x_i}{x_{min}})}$ for each $x_i$ and then solve:
$$\int_\limits{0}^{\infty} ln(\frac{x_i}{x_{min}})\frac{ax_{min}^a}{x_i^{a+1}} dx_i$$
Where the second factor of the integral is the pareto pdf for $x_i$. Can someone help please?
Thanks in advance.