S231 Lecture 4 Cyntha

# N the observed data are now y y1 yn and l

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Unformatted text preview: s, individual p.f' s.) i =1 Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 22 / 45 Likelihood function for random sample & independent experiments Recall if Y1 and Y2 are independent random variables, P (Y1 = y1 , Y2 = y2 ; ) = P (Y1 = y1 ; ) P (Y2 = y2 ; ) Often Y = (Y1 , . . . , Yn ) is a random sample from some process or population, where each Yi has the same probability function (or probability density function) f (y ; ), 2 . n i =1 The observed data are now y = (y1 , . . . , yn ) and L( ) = f (yi ; ) for 2 . (For independent r.v.' joint p.f. is the product of s, individual p.f' s.) Suppose we have independent experiments with observed data y1 and y2 . The "combined" likelihood function L( ) based on y1 and y2 is: L( ) = L1 ( ) L2 ( ) for 2 Jan 2013 22 / 45 where Lj ( ) = P (Yj = yj ; ), for j = 1, 2. Statistics and Actuarial Science () Model Fitting, Estimation and Checking Likelihood for Continuous Distributions For discrete random variables the likelihood function is equal to the probability of observing the data y , that is, L ( ) = L (; y) = P (Y = y; ) for 2 . For continuous distributions, P (Y = y; ) is unsuitable as a de...nition for L ( ) since P (Y = y; ) is always zero. In the continuous case, we de...ne the likelihood function similarly but with the p.f. P (Y = y; ) replaced by the p.d.f. evaluated at the observed values. For independent observations Yi , i = 1, 2, ..., n from the same p.d.f. f (y ; ), the joint p.d.f. of (Y1 , Y2 , ..., Yn ), i =1 f (yi ; ) is used for the likelihood function. n Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 23 / 45 Likelihood for Continuous Distributions For discrete random variables the likelihood function is equal to the probability of observing the data y , that is, L ( ) = L (; y) = P (Y = y; ) for 2 . For continuous distributions, P (Y = y; ) is unsuitable as a de...nition for L ( ) since P (Y = y; ) is always zero. In the continuous case, we de...ne the lik...
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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.

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