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2. Let Y₁,..., Yp be independent random variables such that Y~ N(0, 1). Write Y = (Y₁...., Yp) and 0 = (0₁,...,0p). Let 0 = (Y) = (0₁(Y),..., (Y)) be an estimator of 0, and let g(Y) = (g₁(Y),..., gp (Y)) = - Y. Denote by || - || the Euclidean norm, ||Y||²Y²+...+Y₂. = Suppose that D(Y) = ag(Y)/aY, exists. Then it is known that ‹Â(Y)} = P + 2) + 2 ΣD(Y) + Źl9(11² I=1 i=1 is an unbiased estimator of the risk of , under squared error loss L(0,0) = ||0 − 0||². [You are not required to show this]. (i) The James-Stein estimator is 5Js (Y) = (1-P-2)Y. Show that an unbiased estimator of the risk of djs (Y) is Â(6Js (Y)) = p (p - 2)²/|| Y||². Deduce that Y is inadmissible as an estimator of 0. Is djs (Y) admissible? Justify your answer.

2. Let Y₁,..., Yp be independent random variables such that Y~ N(0, 1). Write Y = (Y₁...., Yp) and 0 = (0₁,...,0p). Let 0 = (Y) = (0₁(Y),..., (Y)) be an estimator of 0, and let g(Y) = (g₁(Y),..., gp (Y)) = - Y. Denote by || - || the Euclidean norm, ||Y||²Y²+...+Y₂. = Suppose that D(Y) = ag(Y)/aY, exists. Then it is known that ‹Â(Y)} = P + 2) + 2 ΣD(Y) + Źl9(11² I=1 i=1 is an unbiased estimator of the risk of , under squared error loss L(0,0) = ||0 − 0||². [You are not required to show this]. (i) The James-Stein estimator is 5Js (Y) = (1-P-2)Y. Show that an unbiased estimator of the risk of djs (Y) is Â(6Js (Y)) = p (p - 2)²/|| Y||². Deduce that Y is inadmissible as an estimator of 0. Is djs (Y) admissible? Justify your answer.

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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2. Let Y,,., Y, be independent random variables such that Y, (Yı., Yp)" and 0 = (0,.,0p)". Let = 0(Y) = (0,(Y),... , @p(Y))" be an estimator of 0, and let g(Y) = (g(Y),... , gp(Y))" = – Y. Denote by || - || the Euclidean norm, ||Y° = Y} + .. + Y. N(8), 1). Write Y = %3D Suppose that D(Y) = @g(Y)/ay, exists. Then it is known that %3D R(Ô(Y)} = +2 D(Y) + É19(Y)² =1 is an unbiased estimator of the risk of 0, under squared error loss L(0, ê) = ||0 – e|P. [You are not required to show this]. %3D (i) The James-Stein estimator is 6.s(Y) = (1 – )Y. _P-2y ||Y? Show that an unbiased estimator of the risk of d Js(Y) is Řlójs(Y)) = p – (p - 2) /||YII°. Deduce that Y is inadmissible as an estimator of 0. Is ô js(Y) admissible? Justify your answer.
Transcribed Image Text:2. Let Y,,., Y, be independent random variables such that Y, (Yı., Yp)" and 0 = (0,.,0p)". Let = 0(Y) = (0,(Y),... , @p(Y))" be an estimator of 0, and let g(Y) = (g(Y),... , gp(Y))" = – Y. Denote by || - || the Euclidean norm, ||Y° = Y} + .. + Y. N(8), 1). Write Y = %3D Suppose that D(Y) = @g(Y)/ay, exists. Then it is known that %3D R(Ô(Y)} = +2 D(Y) + É19(Y)² =1 is an unbiased estimator of the risk of 0, under squared error loss L(0, ê) = ||0 – e|P. [You are not required to show this]. %3D (i) The James-Stein estimator is 6.s(Y) = (1 – )Y. _P-2y ||Y? Show that an unbiased estimator of the risk of d Js(Y) is Řlójs(Y)) = p – (p - 2) /||YII°. Deduce that Y is inadmissible as an estimator of 0. Is ô js(Y) admissible? Justify your answer.
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