Question

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

density function 

f(X; ,0)={ θX^(θ-1), 0<X<1, θ>0; 0, elsewhere

Obtain a maximum likeyhood estimator of θ.

Answer 1

Solution:

Given,

$f(x; θ)=\{ θx^{θ-1}, 0<x<1, θ>0 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0, \ elsewhere$

$L(θ)= \Pi_{i=1}^n \ f(x_i; θ)= \Pi_{i=1}^n \ θx_i^{θ-1} \\=θx_1^{θ-1}.θx_2^{θ-1}.θx_3^{θ-1}....θx_n^{θ-1} \\=θ^n{x_i}^{θ-1}$

$\Rightarrow \log L(θ)=\log (θ^n{x_i}^{θ-1})=\log (θ^n)+\log ({x_i}^{θ-1}) \\=n \logθ+(θ-1)\log x_i$

$\Rightarrow \dfrac{d \ \log L(θ)}{dθ}=\dfrac nθ+\log x_i$

Put $\dfrac{d \ \log L(θ)}{dθ}=0$

$\dfrac nθ+\log x_i=0 \\ \Rightarrow \dfrac nθ=-\log x_i \\ \Rightarrow θ=\dfrac{n}{-\log x_i}=\dfrac{-n}{\log (\dfrac{x_i}{n}.n)}=\dfrac{-n}{(\log \bar x)+(\log n)}$

Thus, MLE $=\hat θ=\dfrac{-n}{(\log \bar x)+(\log n)}$