1. Maximum Likehood Estimator

If f_Y(y;theta) = 2y/(1-theta^2), theta<=y<=1, find the maximum likelihood estimator for theta.

Any help would be great.

2. Originally Posted by eigenvector11
If f_Y(y;theta) = 2y/(1-theta^2), theta<=y<=1, find the maximum likelihood estimator for theta.

Any help would be great.
For an example of what to do, read this: http://www.mathhelpforum.com/math-he...estimator.html

3. Originally Posted by eigenvector11
If f_Y(y;theta) = 2y/(1-theta^2), theta<=y<=1, find the maximum likelihood estimator for theta.

Any help would be great.
In this case you cannot differentiate.
The likelihood function is
$\displaystyle L={2^n\prod_{i=1}^ny_i^2\over (1-\theta)^2} I(\theta \le y_{(1)} \le \cdots \le y_{(n)} \le 1)$
And the largest we can make this is by letting the denominator be as small as possible.
So we want $\displaystyle (1-\theta)^2$ small or $\displaystyle \theta$ big.
BUT by the indicator function the largest $\displaystyle \theta$ can be is
$\displaystyle y_{(1)}$ our smallest order stat.
Hence $\displaystyle \hat\theta_{MLE}=y_{(1)}$