Explanation Let r 1 , r 2 , ... , r r follow a multinomial distribution. In this context, r i is a sum of observations of row i. The likelihood function is obtained as given below: L ( p ) = ( n r 1 r 2 ... r r ) p 1 r 1 p 2 r 2 ... p 2 r r Take natural logarithm on both sides. ln L = ln ( n r 1 + r 1 ) + r 1 ln p 1 + r 2 ln p 2 + ... + r r ln p r It is known that there are n observations; hence, p r = 1 − ∑ i = 1 r − 1 p i r r = n − ∑ i = 1 r − 1 r i Now, ln L can be written as follows: ln L = ln ( n r 1 + r 1 ) + ( ∑ i = 1 r − 1 r i ) ln p i + ( n − ∑ i = 1 r − 1 r i ) ln ( 1 − ∑ i = 1 r − 1 p i ) Here, differentiate ln L with respect to p j . d d p j ln L = r j p j − n − ∑ i = 1 r − 1 r i 1 − ∑ i = 1 r − 1 p i Equate d d p j ln L = 0 , r j p j − n − ∑ i = 1 r − 1 r i 1 − ∑ i = 1 r − 1 p i = 0 r j ( 1 − ∑ i = 1 r − 1 p i ) = p j ( n − ∑ i = 1 r − 1 r i ) [ ∵ j ∈ { 1 , 2 , ..

Mathematical Statistics with Applications

7th Edition

ISBN: 9780495110811

Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer

Publisher: Cengage Learning

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Chapter 14, Problem 39SE

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Verify that MLE of the probability pi for row i is p^i=rin, for i=1,2,...,r.

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Chapter 14 Solutions

Mathematical Statistics with Applications

Ch. 14.3 - Historically, the proportions of all Caucasians in...Ch. 14.3 - Prob. 2E Ch. 14.3 - Prob. 3E Ch. 14.3 - Prob. 4E Ch. 14.3 - Prob. 5E Ch. 14.3 - Prob. 6E Ch. 14.3 - Prob. 7E Ch. 14.3 - Prob. 8E Ch. 14.3 - Prob. 9E Ch. 14.3 - Prob. 10E

Ch. 14.3 - The data in the following table are the frequency...Ch. 14.4 - On the 40th anniversary of President John F....Ch. 14.4 - A study was conducted by Joseph Jacobson and Diane...Ch. 14.4 - Suppose that the entries in a contingency table...Ch. 14.4 - A survey to explore the relationship between...Ch. 14.4 - Prob. 17E Ch. 14.4 - Prob. 18E Ch. 14.4 - The results of a study suggest that the initial...Ch. 14.4 - Refer to Exercise 14.10. Test the hypothesis, at...Ch. 14.4 - An interesting and practical use of the 2 test...Ch. 14.5 - A study to determine the effectiveness of a drug...Ch. 14.5 - The 2 test used in Exercise 14.22 is equivalent to...Ch. 14.5 - How do Americans in the sandwich generation...Ch. 14.5 - Does education really make a difference in how...Ch. 14.5 - A manufacturer of buttons wished to determine...Ch. 14.5 - Traditionally, U.S. labor unions have been content...Ch. 14.5 - Prob. 29E Ch. 14.5 - Refer to Exercise 14.29. Estimate the difference...Ch. 14.5 - A survey was conducted to investigate interest of...Ch. 14 - Prob. 32SE Ch. 14 - A survey was conducted to determine student,...Ch. 14 - How would you rate yourself as a driver? According...Ch. 14 - Prob. 35SE Ch. 14 - Knee injuries are a major problem for athletes in...Ch. 14 - It is often not clear whether all properties of a...Ch. 14 - Prob. 39SE Ch. 14 - Prob. 40SE Ch. 14 - Prob. 41SE Ch. 14 - Prob. 42SE Ch. 14 - Prob. 43SE

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Verify that MLE of the probability p i for row i is p ^ i = r i n , for i = 1 , 2 , ... , r .

Solution Summary: The author explains that MLE of the probability p_i is a sum of observations of row i.

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Verify that MLE of the probability pi for row i is p^i=rin, for i=1,2,...,r.

Chapter 14 Solutions