Maximum Likelihood Estimator

**ft_fan** · January 26th 2010, 07:27 AM

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:

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

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

Show that the maximum likelihood estimator of 1/q is -Sln x $_i$ /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.

Thanks in advance,

ft_fan

**statmajor** · January 26th 2010, 04:53 PM

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

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