Question

Let X₁,X₂,.....,X_n be random sample of size n from a distribution with probability 

density function 

f(X; ,0)={ (θX) to the power (θ-1) ,0<X<1 , θ>0 Obtain a maximum likeyhood

0, elsewhere.

estimator of θ.

Answer 1

$f(X,\theta)=\theta X^{\theta-1}, 0\le X\le1, \theta>0$

$L(\theta)=\prod_{i=1}^nf(X,\theta)$

$=\theta ^n\prod_1^n(Xi)^{\theta-1}$

$\ln L(\theta)=n\ln\theta+(\theta-1)\ln \prod_1^n Xi$

To maximize $\ln L(\theta)$, we equate the first derivative of $\ln L(\theta)$to zero and solve for $\theta$

$\frac{d}{d\theta}\ln L(\theta)=\frac{n}{\theta}+\ln \prod_1^n Xi$

$\frac{n}{\theta}+\ln \prod_1^n Xi=0$

$\frac{n}{\theta}=-\ln \prod_1^n Xi$

$\hat{\theta}=-\frac{n}{\prod_1^nXi}$