S231 Lecture 4 Cyntha

S231 Lecture 4 Cyntha

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Unformatted text preview: ^ Usually obtain by solving the equation d = 0. Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 21 / 45 The Relative Likelihood Function De...nition The Relative likelihood function is R ( ) = L( ) for 2 . ^ L( ) R ( ) has a maximum value equal to one. ^ Log likelihood function: `( ) = log L( ) for 2 . Note that also maximizes `( ). ^ Log relative likelihood function: log R ( ) for 2 . Note that also maximizes this. d` ^ Usually obtain by solving the equation d = 0. y For Binomial model: L( ) = (1 )n y , d` `( ) = y log( ) + (n y ) log(1 ), d = y n y , and d `/d = 0 1 ^ for = y /n so m.l. of is = y /n. Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 21 / 45 L() and log(L()) Maximum of both functions occurs here Instead of maximizing L( ) we can maximize log(L( )) 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 ; ) 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 . 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 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...
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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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