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Cramer-Rao Inequality

  1. Feb 8, 2010 #1
    If [tex]\overline{_}X1[/tex] and [tex]\overline{_}X2[/tex] are the means of independent random samples of size n1 and n2 from a normal population with the mean [tex]\mu[/tex] and [tex]\sigma^2[/tex], show that the variance of the unbiased estimator Var([tex]\omega[/tex][tex]\overline{_}X1[/tex] +(1-[tex]\omega[/tex])[tex]\overline{_}X2[/tex]) is a minimum when [tex]\omega[/tex]= n1 / (n1 + n2).

    My professor gave a hint to find the Var([tex]\omega[/tex][tex]\overline{_}X1[/tex] +(1-[tex]\omega[/tex])[tex]\overline{_}X2[/tex]) and then to minimize with respect to [tex]\omega[/tex] using first and second derivatives.

    I understand to square out the coefficients due to independence but I'm not sure where to begin to find the variances of the means of independent random samples. I feel I should have an understanding to minimize with the Cramer-Rao Inequality. Maybe I'm trying to look too deep into what is being asked?
  2. jcsd
  3. Feb 8, 2010 #2
    Can you show that [tex]\mbox{Var}(\overline{X_1}) = \frac{\sigma^2}{n_1}[/tex]? Hint: write [tex]\overline{X_1}[/tex] as a sum of independent and identically distributed RVs and use [tex]\mbox{Var}(\sum Y_i) = \sum \mbox{Var}(Y_1)[/tex] as well as pulling out a constant becomes multiplying by the square of the constant.
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