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Unformatted text preview: stimates (Gaussian case)
The loglikelihood is `() = `(, ) = n log 1 22 i =1 (yi n )2 (n/2) log(2 ). To maximize `(, ) with respect to both parameters and , solve the two equations ` ` and = = 1 n (yi ) = 0 2 i =1 n 1 n + 3 (yi )2 = 0 i =1 simultaneously. ^ ^ ^ Maximum likelihood estimate of is = (, ), where ^ =
1 n i =1 ^ yi = y , and = n 1 n i =1 (yi n 1/2 y )2 .
Jan 2013 28 / 45 Statistics and Actuarial Science () Model Fitting, Estimation and Checking Likelihoods for Multinomial Models The Multinomial probability function is f (y1 , . . . , yk ; ) =
k k n! y i i for yi = 0, 1, . . . where yi = n y1 ! . . . yk ! i =1 i =1 If the i ' are to be estimated from data involving n "trials", of which yi s resulted in outcome i, i = 1, . . . , k, then it seems obvious that ^ i = yi /n for i = 1, . . . , k would be a sensible estimate. This can also be shown to be the m.l. estimate for = ( 1 , . . . , k ). Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 29 / 45 University and college teachers earn too much
60% 50% 40% 30% 20% 10% 0% Agree
Nov. 09 Disagree
Nov. 10 Neutral
Nov. 11 Don't know
Nov. 12 Likelihoods for Multinomial Models 2
Example: Consider the Harris/Decima example for the 4 responses: "Agree", "Disagree", Neutral", and Don' know" in November 2012. Let t 1 , 2 , 3 , 4 be the fraction of a population that "Agree", "Disagree", Neutral", and Don' know", respectively. In the sample of 2000 persons t the numbers who "Agree", "Disagree", Neutral", and Don' know" were t y1 = 640, y2 = 860, y3 = 340 and y4 = 160 (note that y1 + y2 + y3 + y4 = 2000). Let the random variables Y1 , Y2 , Y3 , Y4 represent the number who "Agree", "Disagree", Neutral", and Don' know" that we might get t in a random sample of size n = 2000. Then Y1 , Y2 , Y3 , Y4 follow a Multinomial(2000; 1 , 2 , 3 , 4 ). Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 31 / 45 Likelihoods for Multinomial Models 2
Example: Consider the Harris/Decima example for the 4 responses: "Agree", "Disagree", Neutral", and Don' know" in November 2012. Let t 1 , 2 , 3 , 4 be the fraction of a population that "Agree", "Disagree", Neutral", and Don' know", respectively. In the sample of 2000 persons t the numbers who "Agree", "Disagree", Neutral", and Don' know" were t y1 = 640, y2 = 860, y3 = 340 and y4 = 160 (note that y1 + y2 + y3 + y4 = 2000). Let the random variables Y1 , Y2 , Y3 , Y4 represent the number who "Agree", "Disagree", Neutral", and Don' know" that we might get t in a random sample of size n = 2000. Then Y1 , Y2 , Y3 , Y4 follow a Multinomial(2000; 1 , 2 , 3 , 4 ). The maximum likelihood estimates of 1 , 2 , 3 , 4 from the observed 640 860 340 ^ ^ ^ data are 1 = 2000 = 0.32, 2 = 2000 = 0.43, 3 = 2000 = 0.17, ^ 4 =
160 2000 ^ = 0.08 (check that i = 1).
i =1
Model Fitting, Estimation and Checking Jan 2013 31 / 45 4 Statistics and Actuarial Science ()...
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This note was uploaded on 04/17/2013 for the course STAT 231 taught by Professor Cantremember during the Winter '08 term at Waterloo.
 Winter '08
 CANTREMEMBER
 Statistics

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