Question

4. For the distribution f(z; a) = (a +1)xº, 0<x<1, what is the MLE of a based on a random sample X1, X2, ..., Xn?

Answer 1

Given,

$f(x)=(\alpha+1)x^\alpha\, , \qquad \qquad 0\leq x\leq 1$

Likelihood function is given by,

L = II f(x) = (a +1)" II 22 isi

Taking logarithm on both sides, we get,

$\log L=n\log(\alpha+1)+\alpha\sum_{i=1}^{n}\log(x_i)$

Differentiating with respect to $\alpha$ and equate to zero. Then solve for $\alpha$ , we get the mle $\hat{\alpha}$ .

$\frac{\partial \log L}{\partial \alpha}=0\\ \begin{align*} \frac{n}{\alpha+1}+\sum_{i=1}^{n}\log(x_i)&=0\\ \frac{n}{\alpha+1}&=-\sum_{i=1}^{n}\log(x_i)\\ \alpha+1&=\frac{-n}{\sum_{i=1}^{n}\log(x_i)} \intertext{Therefore} \hat{\alpha}&=\frac{-n}{\sum_{i=1}^{n}\log(x_i)}-1 \end{align*}$