Mathematical Statistics with Applications
Mathematical Statistics with Applications
7th Edition
ISBN: 9780495110811
Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer
Publisher: Cengage Learning
Question
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Chapter 9.6, Problem 69E
To determine

Provide an estimator of θ by the method of moments.

Prove that the estimator is consistent.

Explain whether the estimator is a function of the sufficient statistic i=1nln(Yi), that can be obtained from the factorization criterion.

Also provide the implication.

Expert Solution & Answer
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Answer to Problem 69E

The method of moments estimator of θ is 2Y¯11Y¯.

Explanation of Solution

Calculation:

The expectation of the given random variable Y is obtained as follows:

E(Y)=01y(θ+1)yθdy=01(θ+1)yθ+1dy=(θ+1)[yθ+2θ+2]01=θ+1θ+2

Consider that the method of moments estimator of θ is θ^.

Now, to obtained the method of moments estimator of θ, it is needed to equate the expectation with the sample mean. That is,

θ^+1θ^+2=Y¯θ^+1=Y¯θ^+2Y¯θ^(1Y¯)=2Y¯1θ^=2Y¯11Y¯

Hence, the method of moments estimator of θ is 2Y¯11Y¯.

Now consider,

E(Y¯)=E(θ+1θ+2)=θ+1θ+2

Thus, Y¯ is an unbiased estimator of θ+1θ+2.

Now,

E(Y2)=01y2(θ+1)yθdy=01(θ+1)yθ+2dy=(θ+1)[yθ+3θ+3]01=θ+1θ+3

The variance of Y is given as follows:

V(Y)=E(Y2)[E(Y)]2=θ+1θ+3(θ+1)2(θ+2)2=(θ+1)(θ+2)2(θ+3)(θ+1)2(θ+2)2(θ+3)=(θ+1)(θ2+2θ+4)(θ+3)(θ2+2θ+1)(θ+2)2(θ+3)=θ3+2θ2+4θ+θ2+2θ+4θ32θ2θ3θ26θ3(θ+2)2(θ+3)=12θ2θ(θ+2)2(θ+3)

The variance of Y¯ is given as follows:

V(Y¯)=V(1ni=1nY)=1n2V(i=1nY)=1n2i=1nV(Y)                                      ( Yi's are independent)=1n2[12θ2θ(θ+2)2(θ+3)+...+12θ2θ(θ+2)2(θ+3)(n times)]   =nn2(12θ2θ(θ+2)2(θ+3))=12θ2θn(θ+2)2(θ+3)limnV(Y¯)=limn12θ2θn(θ+2)2(θ+3)=0

Thus, the estimator Y¯ is a consistent estimator of θ+1θ+2.

Now, using Law of large numbers it can be aid that Y¯ converges in probability to θ+1θ+2.

That is,

θ^=2Y¯11Y¯θ^2(θ+1θ+2)11(θ+1θ+2)θ^θ

Hence, θ^ converges to θ.

The likelihood function of α can be written as follows:

L(α)=(θ+1)n(i=1nyi)θ=h(y)g(i=1nyi,θ)

Where h(y)=1 and g(i=1nyi,θ)=(θ+1)n(i=1nyi)θ.

By Theorem 9.4 (Factorization theorem), it can be said that, i=1nYi is sufficient for θ.

Hence, the estimator is estimator is not a function of the sufficient statistic i=1nln(Yi). Moreover, it is the function of i=1nYi.

Therefore, it implies that it is not a minimum variance unbiased estimator.

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Chapter 9 Solutions

Mathematical Statistics with Applications

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