S231 Lecture 4 Cyntha

N the observed data are now y y1 yn and l

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online