# Maximum Likelihood Estimator

• Jan 26th 2010, 07:27 AM
ft_fan
Maximum Likelihood Estimator
Hi all,

I have been going through a practice paper for an exam I have on Friday and have found a question that I don't think we've been taught. Could anyone shed some light on the following question:

Quote:

Let x1,x2,....,xnbe a random sample from a population whose probability density is given by

$\displaystyle f(x) = \theta x^ {\theta - 1}, 0<x<1, \theta>1$

Show that the maximum likelihood estimator of 1/q is -Sln x$\displaystyle _i$/n, and that this estimator is unbiased.
I'm not asking for the question to be solved, but if someone point me in the right direction, I'd be very grateful.

1.$\displaystyle L(X_1,..X_n; \theta) = \prod f(x_i ; \theta)L(X_1,..X_2; \theta) = \prod f(x_i ; \theta)$
2. $\displaystyle l(\theta) = ln L(X_1,...,X_n ; \theta)$ (take the natural logarithm of the likelihood function)
3.Take the partial derivative of $\displaystyle l(\theta)$ (also called the score function) and set it to 0. Now, solve for theta hat.