Transcription of Generalized Linear Model Theory - Princeton University
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Generalized Linear Model Theory - Princeton University
Appendix BGeneralized Linear ModelTheoryWe describe the Generalized Linear Model as formulated by Nelder and Wed-derburn (1972), and discuss estimation of the parameters and tests of The ModelLety1, .. , yndenotenindependent observations on a response. We treatyias a realization of a random variableYi. In the general Linear Model weassume thatYihas a normal distribution with mean iand variance 2Yi N( i, 2),and we further assume that the expected value iis a Linear function ofppredictors that take valuesx i= (xi1, .. , xip) for thei-th case, so that i=x i ,where is a vector of unknown will generalize this in two steps, dealing with the stochastic andsystematic components of the Rodr guez. Revised November 20012 APPENDIX B. Generalized Linear Model The Exponential FamilyWe will assume that the observations come from a distribution in the expo-nential family with probability density functionf(yi) = exp{yi i b( i)ai( )+c(yi, )}.( )Here iand are parameters andai( ),b( i) andc(yi, ) are known func-tions.
B.1. THE MODEL 3 Try to generalize this result to the case where Y i has a normal distribution with mean µ i and variance σ2/n i for known constants n i, as would be the case if the Y i represented sample means. Example: In Problem Set 1 …
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