S231 Lecture 4 Cyntha

Then y1 y2 y3 y4 follow a multinomial2000 1 2

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Unformatted text preview: ma 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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