Unformatted text preview: γ . An unbiased estimator of γ is ˆ γ = n1 ∑ n i =1 ln(1 + X i ) . (You do not have to show that it is unbiased.) Does the variance of this estimator achieve the CRLB? 5. Required for 5520 students only The quantity U ( v X, θ ) := ∂ ∂θ ln f ( v X ; θ ) given in the CRLB is known as the “score statistic”. (a) Show that the variance of the score statistic is the Fisher information. (b) Suppose there exists an unbiased estimator T ( v X ) of τ ( θ ). Show that T ( v X ) attains the CRLB if and only if the score statistic can be expressed in the form U ( v X ; θ ) = g ( θ )[ T ( v X )τ ( θ )] for some function g ( θ ). ( Hint: Check out the proof of the CRLB on the website. )...
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 Fall '11
 Manuel
 Normal Distribution, Variance, Probability theory, CRLB, cram´rrao lower bound

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