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Get Full Access to Mathematical Statistics With Applications - 7 Edition - Chapter 9 - Problem 66e
Get Full Access to Mathematical Statistics With Applications - 7 Edition - Chapter 9 - Problem 66e

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# The likelihood function L(y1, y2, . . . , yn | ?) takes on

ISBN: 9780495110811 47

## Solution for problem 66E Chapter 9

Mathematical Statistics with Applications | 7th Edition

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Problem 66E

Problem 66E

The likelihood function L(y1, y2, . . . , yn | θ) takes on different values depending on the arguments (y1, y2 , . . . , yn ). A method for deriving a minimal sufficient statistic developed by Lehmann and Scheff´e uses the ratio of the likelihoods evaluated at two points, (x1, x2, . . . , xn) and (y1, y2, . . . , yn ):

Many times it is possible to find a function g(x1, x2, . . . , xn ) such that this ratio is free of the unknown parameter θ if and only if g(x1, x2, . . . , xn ) = g(y1, y2, . . . , yn ). If such a function g can be found, then g(Y1, Y2, . . . , Yn ) is a minimal sufficient statistic for θ .

a Let Y1, Y2, . . . , Yn be a random sample from a Bernoulli distribution (see Example 9.6 and Exercise 9.65) with p unknown.

i Show that

ii Argue that for this ratio to be independent of p, we must have

iii Using the method of Lehmann and Scheff´e, what is a minimal sufficient statistic for p? How does this sufficient statistic compare to the sufficient statistic derived in Example 9.6 by using the factorization criterion?

b Consider the Weibull density discussed in Example 9.7.

i Show that

Reference

In this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let Y1, Y2, . . . , Ynbe independent Bernoulli random variables with

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##### ISBN: 9780495110811

Mathematical Statistics with Applications was written by and is associated to the ISBN: 9780495110811. This full solution covers the following key subjects: sufficient, Statistic, function, Example, ratio. This expansive textbook survival guide covers 32 chapters, and 3350 solutions. The full step-by-step solution to problem: 66E from chapter: 9 was answered by , our top Statistics solution expert on 07/18/17, 08:07AM. This textbook survival guide was created for the textbook: Mathematical Statistics with Applications , edition: 7. Since the solution to 66E from 9 chapter was answered, more than 282 students have viewed the full step-by-step answer. The answer to “The likelihood function L(y1, y2, . . . , yn | ?) takes on different values depending on the arguments (y1, y2 , . . . , yn ). A method for deriving a minimal sufficient statistic developed by Lehmann and Scheff´e uses the ratio of the likelihoods evaluated at two points, (x1, x2, . . . , xn) and (y1, y2, . . . , yn ): Many times it is possible to find a function g(x1, x2, . . . , xn ) such that this ratio is free of the unknown parameter ? if and only if g(x1, x2, . . . , xn ) = g(y1, y2, . . . , yn ). If such a function g can be found, then g(Y1, Y2, . . . , Yn ) is a minimal sufficient statistic for ? .a Let Y1, Y2, . . . , Yn be a random sample from a Bernoulli distribution (see Example 9.6 and Exercise 9.65) with p unknown.i Show that ii Argue that for this ratio to be independent of p, we must have iii Using the method of Lehmann and Scheff´e, what is a minimal sufficient statistic for p? How does this sufficient statistic compare to the sufficient statistic derived in Example 9.6 by using the factorization criterion?b Consider the Weibull density discussed in Example 9.7.i Show that ReferenceIn this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let Y1, Y2, . . . , Ynbe independent Bernoulli random variables with” is broken down into a number of easy to follow steps, and 253 words.

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The likelihood function L(y1, y2, . . . , yn | ?) takes on