us be given p 23 independent normally distributed random variables i~ N(μi, 1), i = 1,..., p. Let Y = (Y₁,..., Yp)' and μ = (₁, ...,Hp)'. Let The loss of a decision rule 8(Y) for estimating the parameter vector µ be (µ, 8(Y)) = (§(Y) — µ)'(8(Y) − µ) = Σ1 (8(Y)i − µi)². - a) Determine the risk of the estimator ₁ (Y) = Y. b) Show that the risk of the estimator is given as 8₂ (Y)= (1- (p=²))Y, Z=Y'Y, Z - p(μ, 8₂) = p (p - 2)² E(1/Z). - [Hint: Use the identity E[(Yu)'Y/Z]= (p2) E(1/Z).] c) Compare both risks. What can be said?
us be given p 23 independent normally distributed random variables i~ N(μi, 1), i = 1,..., p. Let Y = (Y₁,..., Yp)' and μ = (₁, ...,Hp)'. Let The loss of a decision rule 8(Y) for estimating the parameter vector µ be (µ, 8(Y)) = (§(Y) — µ)'(8(Y) − µ) = Σ1 (8(Y)i − µi)². - a) Determine the risk of the estimator ₁ (Y) = Y. b) Show that the risk of the estimator is given as 8₂ (Y)= (1- (p=²))Y, Z=Y'Y, Z - p(μ, 8₂) = p (p - 2)² E(1/Z). - [Hint: Use the identity E[(Yu)'Y/Z]= (p2) E(1/Z).] c) Compare both risks. What can be said?
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![1.1. Let us be given p ≥ 3 independent normally distributed random variables
p)'. Let
Yi~ N(₁, 1), i = 1,..., p. Let Y = (Y₁,..., Yp)' and μ = (₁,
(µi,
-
the loss of a decision rule ¿(Y) for estimating the parameter vector µ be
L(µ,8(Y)) = (8(Y) − µ)'(8(Y) − µ) = Σi_₁ (8(Y); − µi)².
-
-
i=1
(a) Determine the risk of the estimator ₁ (Y) = Y.
(b) Show that the risk of the estimator
is given as
8₂ (Y) = (1-
(p − 2)
Z
-)Y, Z=Y'Y,
p(µ, 8₂) = p – (p − 2)² E(1/Z) .
[Hint: Use the identity E[(Yu)'Y/Z]= (p-2) E(1/Z).]
(c) Compare both risks. What can be said?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ea1f414-2cfa-4d12-9b16-9c9fcbba0754%2Fade43138-88a0-4046-9f05-0284cd11c1f1%2F5y7uvy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.1. Let us be given p ≥ 3 independent normally distributed random variables
p)'. Let
Yi~ N(₁, 1), i = 1,..., p. Let Y = (Y₁,..., Yp)' and μ = (₁,
(µi,
-
the loss of a decision rule ¿(Y) for estimating the parameter vector µ be
L(µ,8(Y)) = (8(Y) − µ)'(8(Y) − µ) = Σi_₁ (8(Y); − µi)².
-
-
i=1
(a) Determine the risk of the estimator ₁ (Y) = Y.
(b) Show that the risk of the estimator
is given as
8₂ (Y) = (1-
(p − 2)
Z
-)Y, Z=Y'Y,
p(µ, 8₂) = p – (p − 2)² E(1/Z) .
[Hint: Use the identity E[(Yu)'Y/Z]= (p-2) E(1/Z).]
(c) Compare both risks. What can be said?
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