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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ffb3b62-638a-4739-85e1-a30a305f8e83%2F3fddf1e4-3852-4405-b10d-58fb74237fa0%2Fmn0sqdi_processed.png&w=3840&q=75)
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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