Answer to Question #166025 in Statistics and Probability for Chang

Question #166025
  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


1
Expert's answer
2021-02-24T06:04:00-0500

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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