
MLE and its properties
Hi guys,
May I obtain some feedback on my attempts on this question:
The question is as follows:
http://img199.imageshack.us/img199/1504/83863226.png
1) The first part required me to find an estimate for theta using MLE where I first defined the log likelihood function as:
http://img828.imageshack.us/img828/6908/31056477.png
I then differentiated this with respect to theta and equated it to 0 to solve for an estimator of theta where I obtain:
$\displaystyle \hat{\theta} = \frac{\sum y}{n\alpha}$
2) The second part requires that I estimate the variance of the MLE I have just derived so I took the following approach:
$\displaystyle var[\hat{\theta}] = var\left[ \frac{\sum y}{n\alpha} \right]$
I worked on the RHS using variance properties $\displaystyle var[Y] = b^2 var[X]$ to obtain
$\displaystyle \left( \frac{1}{n\alpha} \right)^2 var[\sum y]$
The question originally gave $\displaystyle var[y] = \alpha \theta^2$ so substituting this in gives:
$\displaystyle \frac{\theta^2}{n \alpha}$
Finally I substitute the estimate for theta I derived in part 1) to obtain an answer of:
$\displaystyle \frac{(\sum y)^2}{n^3 \alpha^3}$
Thank you in advance for any feedback given
Lin