Question

Problem 4 Let Yı, Y2, ..., Y, denote a random sample from the probability density function (0 + 1)ye f(0) = 0 <y <1,0 > -1 elsewhere Find the MLE for .

Answer 1

Answer:

Given Data

Given pdf of Y is

$f(y|\theta)=\left\{\begin{matrix} (\theta+1)y^{\theta} &, 0<y<1,\theta>-1\\ 0 &,elsewhere \end{matrix}\right.$Likelihood function is

$\begin{align*} L(y_{i}|\theta) &=\prod_{i=1}^{n}f(y_{i}|\theta) \\ &= \prod_{i=1}^{n}(\theta+1)y_{i}^{\theta}\\ &= (\theta +1)^{n}\left ( \prod_{i=1}^{n}y_{i} \right )^{\theta} \end{align*}$

Taking log on both sides ,

$\therefore$ Log likelihood of function is

$\begin{align*} l(\theta) &=logL(y_{i}|\theta) \\ &= n\,log(\theta+1)+\theta \,\,log\left ( \prod_{i=1}^{n}y_{i} \right )\\ \therefore \,\,put,\,\,\frac{dl(\theta)}{d\theta} &=0 \\ \frac{n}{\theta+1}+log\left ( \prod_{i=1}^{n}y_{i} \right )&= 0\\ n+(\theta+1)log\left ( \prod_{i=1}^{n}y_{i} \right )&=0 \\ (\theta+1)log \left ( \prod_{i=1}^{n}y_{i} \right )&=-n \\ \theta+1&=\frac{-n}{log\left ( \prod_{i=1}^{n}y_{i} \right )} \\ MLE\,\,of\,\,\theta\,\,is\\ \theta &=-\left (1+\frac{n}{log\left ( \prod_{i=1}^{n}y_{i} \right )} \right ) \end{align*}$

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