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

Ch. 9.3 - Applet Exercise Refer to Exercises 9.9 and 9.10....Ch. 9.3 - Applet Exercise Refer to Exercise 9.11. What...Ch. 9.3 - Applet Exercise Refer to Exercises 9.99.12. Access...Ch. 9.3 - Applet Exercise Refer to Exercise 9.13. Scroll...Ch. 9.3 - Refer to Exercise 9.3. Show that both 1 and 2 are...Ch. 9.3 - Refer to Exercise 9.5. Is 22 a consistent...Ch. 9.3 - Suppose that X1, X2,, Xn and Y1, Y2,,Yn are...Ch. 9.3 - In Exercise 9.17, suppose that the populations are...Ch. 9.3 - Let Y1, Y2,,Yn denote a random sample from the...Ch. 9.3 - If Y has a binomial distribution with n trials and...Ch. 9.3 - Let Y1, Y2,, Yn be a random sample of size n from...Ch. 9.3 - Refer to Exercise 9.21. Suppose that Y1, Y2,, Yn...Ch. 9.3 - Refer to Exercise 9.21. Suppose that Y1, Y2,, Yn...Ch. 9.3 - Let Y1, Y2, Y3, Yn be independent standard normal...Ch. 9.3 - Suppose that Y1, Y2, , Yn denote a random sample...Ch. 9.3 - Prob. 26ECh. 9.3 - Use the method described in Exercise 9.26 to show...Ch. 9.3 - Let Y1, Y2, , Yn denote a random sample of size n...Ch. 9.3 - Let Y1, Y2, , Yn denote a random sample of size n...Ch. 9.3 - Let Y1, Y2, , Yn be independent random variables,...Ch. 9.3 - Prob. 31ECh. 9.3 - Let Y1, Y2, , Yn denote a random sample from the...Ch. 9.3 - An experimenter wishes to compare the numbers of...Ch. 9.3 - Prob. 34ECh. 9.3 - Let Y1, Y2, be a sequence of random variables with...Ch. 9.3 - Suppose that Y has a binomial distribution based...Ch. 9.4 - Prob. 37ECh. 9.4 - Let Y1, Y2, , Yn denote a random sample from a...Ch. 9.4 - Let Y1, Y2, , Yn denote a random sample from a...Ch. 9.4 - Prob. 40ECh. 9.4 - Let Y1, Y2, , Yn denote a random sample from a...Ch. 9.4 - If Y1, Y2, , Yn denote a random sample from a...Ch. 9.4 - Prob. 43ECh. 9.4 - Let Y1, Y2, , Yn denote independent and...Ch. 9.4 - Suppose that Y1, Y2, , Yn is a random sample from...Ch. 9.4 - If Y1, Y2,, Yn denote a random sample from an...Ch. 9.4 - Refer to Exercise 9.43. If is known, show that...Ch. 9.4 - Refer to Exercise 9.44. If is known, show that...Ch. 9.4 - Let Y1, Y2, . . . , Yn denote a random sample from...Ch. 9.4 - Let Y1, Y2, . . . , Yn denote a random sample from...Ch. 9.4 - Prob. 51ECh. 9.4 - Prob. 52ECh. 9.4 - Prob. 53ECh. 9.4 - Prob. 54ECh. 9.4 - Let Y1, Y2, . . . , Yn denote independent and...Ch. 9.5 - Refer to Exercise 9.38(b). Find an MVUE of 2. 9.38...Ch. 9.5 - Refer to Exercise 9.18. Is the estimator of 2...Ch. 9.5 - Refer to Exercise 9.40. Use i=1nYi2 to find an...Ch. 9.5 - The number of breakdowns Y per day for a certain...Ch. 9.5 - Prob. 60ECh. 9.5 - Refer to Exercise 9.49. Use Y(n) to find an MVUE...Ch. 9.5 - Refer to Exercise 9.51. Find a function of Y(1)...Ch. 9.5 - Prob. 63ECh. 9.5 - Let Y1, Y2, , Yn be a random sample from a normal...Ch. 9.5 - In this exercise, we illustrate the direct use of...Ch. 9.5 - The likelihood function L(y1,y2,,yn|) takes on...Ch. 9.5 - Refer to Exercise 9.66. Suppose that a sample of...Ch. 9.5 - Prob. 68ECh. 9.6 - Prob. 69ECh. 9.6 - Suppose that Y1, Y2, , Yn constitute a random...Ch. 9.6 - If Y1, Y2, , Yn denote a random sample from the...Ch. 9.6 - If Y1, Y2, , Yn denote a random sample from the...Ch. 9.6 - An urn contains black balls and N white balls....Ch. 9.6 - Let Y1, Y2,, Yn constitute a random sample from...Ch. 9.6 - Prob. 75ECh. 9.6 - Let X1, X2, X3, be independent Bernoulli random...Ch. 9.6 - Let Y1, Y2,, Yn denote independent and identically...Ch. 9.6 - Let Y1, Y2,, Yn denote independent and identically...Ch. 9.6 - Let Y1, Y2,, Yn denote independent and identically...Ch. 9.7 - Suppose that Y1, Y2,, Yn denote a random sample...Ch. 9.7 - Suppose that Y1, Y2, , Yn denote a random sample...Ch. 9.7 - Prob. 82ECh. 9.7 - Suppose that Y1, Y2, , Yn constitute a random...Ch. 9.7 - Prob. 84ECh. 9.7 - Let Y1, Y2,, Yn denote a random sample from the...Ch. 9.7 - Suppose that X1, X2, , Xm, representing yields per...Ch. 9.7 - A random sample of 100 voters selected from a...Ch. 9.7 - Prob. 88ECh. 9.7 - It is known that the probability p of tossing...Ch. 9.7 - A random sample of 100 men produced a total of 25...Ch. 9.7 - Find the MLE of based on a random sample of size...Ch. 9.7 - Prob. 92ECh. 9.7 - Prob. 93ECh. 9.7 - Suppose that is the MLE for a parameter . Let t()...Ch. 9.7 - A random sample of n items is selected from the...Ch. 9.7 - Consider a random sample of size n from a normal...Ch. 9.7 - The geometric probability mass function is given...Ch. 9.8 - Refer to Exercise 9.97. What is the approximate...Ch. 9.8 - Consider the distribution discussed in Example...Ch. 9.8 - Suppose that Y1, Y2, . . . , Yn constitute a...Ch. 9.8 - Let Y1, Y2, . . . , Yn denote a random sample of...Ch. 9.8 - Refer to Exercises 9.97 and 9.98. If a sample of...Ch. 9 - Prob. 103SECh. 9 - Prob. 104SECh. 9 - Refer to Exercise 9.38(b). Under the conditions...Ch. 9 - Prob. 106SECh. 9 - Suppose that a random sample of length-of-life...Ch. 9 - The MLE obtained in Exercise 9.107 is a function...Ch. 9 - Prob. 109SECh. 9 - Refer to Exercise 9.109. a Find the MLE N2 of N. b...Ch. 9 - Refer to Exercise 9.110. Suppose that enemy tanks...Ch. 9 - Let Y1, Y2, . . . , Yn denote a random sample from...
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