1. Consider a Gaussian statistical model X₁,..., Xn~ N(0, 0), with unknown > 0. Note that Var(X) = 0 and Var (X²) = 202. To simplify the notation, define X = 1X²/n. Prove that = X is the maximum likelihood estimator for 0, and verify that it (a) is unbiased. (b) Prove that the expected Fisher information for is equal to n/(202), and check if the maximum likelihood estimator for attains the Cramer-Rao lower bound. (c) Prove that the maximum likelihood estimator for is based on a minimal sufficient statistic. (d) Identify the parameters of a Gaussian density which is approximately propor- tional to the likelihood function of 0, in a neighbourhood of its maximum likelihood estimator.
1. Consider a Gaussian statistical model X₁,..., Xn~ N(0, 0), with unknown > 0. Note that Var(X) = 0 and Var (X²) = 202. To simplify the notation, define X = 1X²/n. Prove that = X is the maximum likelihood estimator for 0, and verify that it (a) is unbiased. (b) Prove that the expected Fisher information for is equal to n/(202), and check if the maximum likelihood estimator for attains the Cramer-Rao lower bound. (c) Prove that the maximum likelihood estimator for is based on a minimal sufficient statistic. (d) Identify the parameters of a Gaussian density which is approximately propor- tional to the likelihood function of 0, in a neighbourhood of its maximum likelihood estimator.
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Author:Amos Gilat
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
Transcribed Image Text:1. Consider a Gaussian statistical model X₁, ..., Xn ~ N(0, 0), with unknown > 0. Note that
Var (X) = 0 and Var (X²) = 20². To simplify the notation, define X = E=₁ X²/n.
Prove that = X is the maximum likelihood estimator for 0, and verify that it
(a)
is unbiased.
(b)
Prove that the expected Fisher information for is equal to n/(20²), and check
if the maximum likelihood estimator for attains the Cramer-Rao lower bound.
(c)
Prove that the maximum likelihood estimator for is based on a minimal
sufficient statistic.
(d)
Identify the parameters of a Gaussian density which is approximately propor-
tional to the likelihood function of 0, in a neighbourhood of its maximum likelihood
estimator.
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