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Suppose X1, X2, X3, X4 be i.i.d. normal random variables with mean 0 and variance o, where o is the unknown parameter. Consider the following estimators: T1 X- X2 + X4, T2 (X? + X3 + X3), T3 E X7, T4 = L(X-X), and T, = |X1- X2l. %3D %3D (a) Is T unbiased for o?, for i = 1,2,3, 4? (b) Among the estimators T,...,T for o, which has the smallest MSE? (c) Is T, unbiased for o? If not, find a constant k so that kT, is unbiased for o. Evaluate the MSE of Ts.

Suppose X1, X2, X3, X4 be i.i.d. normal random variables with mean 0 and variance o, where o is the unknown parameter. Consider the following estimators: T1 X- X2 + X4, T2 (X? + X3 + X3), T3 E X7, T4 = L(X-X), and T, = |X1- X2l. %3D %3D (a) Is T unbiased for o?, for i = 1,2,3, 4? (b) Among the estimators T,...,T for o, which has the smallest MSE? (c) Is T, unbiased for o? If not, find a constant k so that kT, is unbiased for o. Evaluate the MSE of Ts.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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Suppose X1, X2, X3, X4 be i.i.d. normal random variables with mean 0 and variance sigma^2, where sigma^2 is the unknown parameter.

Consider the following estimators: T1 = X1 - X2 + X4, T2 =  1/3(X1 + X1 + X4), T3... T4... T5 = 1/2|X1 - X2|

(a) Is T1 unbiased for sigma^2, for i = 1,2,3,4

(b) Among the estimators T1,...,T4 for sigma^2, which has the smallest MSE?

(c) Is T5 unbiased for sigma? If not, find a constant k so that k*T5 is unbiased for sigma^2. Evaluate the MSE of T5.

Suppose X1, X2, X3, X4 be i.i.d. normal random variables with mean 0 and variance o?, where o is the unknown parameter. Consider the following estimators: T1 X- X2 + X4, T2 = 3(X? + X3 + X3), T3 Ex, T L(X-X), and Ts X1- Xal. %3D (a) Is T, unbiased for o?, for i 1,2,3, 4? (b) Among the estimators T,...,T4 for o?, which has the smallest MSE? (c) Is T, unbiased for o? If not, find a constant k so that KT, is unbiased for o. Evaluate the MSE of T3. %D
Transcribed Image Text:Suppose X1, X2, X3, X4 be i.i.d. normal random variables with mean 0 and variance o?, where o is the unknown parameter. Consider the following estimators: T1 X- X2 + X4, T2 = 3(X? + X3 + X3), T3 Ex, T L(X-X), and Ts X1- Xal. %3D (a) Is T, unbiased for o?, for i 1,2,3, 4? (b) Among the estimators T,...,T4 for o?, which has the smallest MSE? (c) Is T, unbiased for o? If not, find a constant k so that KT, is unbiased for o. Evaluate the MSE of T3. %D
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