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