Answer to Question #166025 in Statistics and Probability for Chang

Question

1. let x1 x2 xn be a random sample from a population with pdf ... f(x theta) =theta. e power - theta .x

find cramer roa lower bound for variance of unbiased estimator of theta

Answer 1

Given Probability density Function is-

$F(X,\theta)=\theta e ^{-\theta}$

Let $l(X,\theta)$ be the natural logrithm of $f(x,\theta)$

$l=log(\theta e^{-\theta})=log\theta+loge^{-\theta}=log\theta-\theta$

Variance of unbiased estimator of \theta

$V(\hat{\theta})\ge \dfrac{1}{I(\theta)}~~~~~~~-(1)$

Where, the Fisher information ${\displaystyle I(\theta )}$  is defined by

$I(\theta)=-E[\dfrac{d^2l(X,\theta)}{d\theta^2}]$

Where, $E=$ Expected value of samples

$I(\theta)=-E(-\dfrac{1}{\theta^2})$

$I(\theta)=\dfrac{E}{\theta^2}$

From eqs.(1) we have

$V(\hat{\theta})\ge \dfrac{1}{I(\theta)}$

$V(\hat{\theta})\ge\dfrac{\theta^2}{E}$