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Let p denote a loss function, and let X₁,..., X, id P. Let denote the M-estimator for some unknown parameter * = argmin ¹μERE [p (X₁, µ)] = R associated with P. (Here we are assuming that μ* is a one-dimensional parameter.) Consider the following functions I(u) •[0²P (X, 11]

Let p denote a loss function, and let X₁,..., X, id P. Let denote the M-estimator for some unknown parameter * = argmin ¹μERE [p (X₁, µ)] = R associated with P. (Here we are assuming that μ* is a one-dimensional parameter.) Consider the following functions I(u) •[0²P (X, 11]

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Concept Check: M-estimators vs. Maximum Likelihood Estimation iid Let p denote a loss function, and let X₁,..., Xn P. Let denote the M-estimator for some unknown parameter μ* = argmin μERE [p(X₁, μ)] € R associated with P. (Here we are assuming that μ* is a one-dimensional parameter.) Consider the following functions J (μ) = E (μ² (X1,μ) ap Əμ Which of the following statements are true? (Choose all that apply.) K(μ) = Var -(X₁₂μ) It is always true that J (u) = K (μ). J(μ) = K (μ) when p is the negative log-likelihood- in this case, both of these functions are equal to the Fisher information. Under some technical conditions, the functions J (u) and K (μ) determine the asymptotic variance of the M-estimator μ.
Transcribed Image Text:Concept Check: M-estimators vs. Maximum Likelihood Estimation iid Let p denote a loss function, and let X₁,..., Xn P. Let denote the M-estimator for some unknown parameter μ* = argmin μERE [p(X₁, μ)] € R associated with P. (Here we are assuming that μ* is a one-dimensional parameter.) Consider the following functions J (μ) = E (μ² (X1,μ) ap Əμ Which of the following statements are true? (Choose all that apply.) K(μ) = Var -(X₁₂μ) It is always true that J (u) = K (μ). J(μ) = K (μ) when p is the negative log-likelihood- in this case, both of these functions are equal to the Fisher information. Under some technical conditions, the functions J (u) and K (μ) determine the asymptotic variance of the M-estimator μ.
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