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Chapter 3 Notes - Exponential family of distributions...

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Exponential family of distributions Probability distribution on Y that has the form, f ( y | θ ) = exp [ a ( y ) b ( θ ) + c ( θ ) d ( y )] a ( · ) , d ( · ) are functions of y . b ( · ) , c ( · ) are functions of θ . b ( θ ) is called the natural parameter . In fact, E ( a ( Y )) = - c 0 ( θ ) b 0 ( θ ) ; Var ( a ( Y )) = b 00 ( θ ) c 0 ( θ ) - c 00 ( θ ) b 0 ( θ ) [ b 0 ( θ )] 3 Normal, Poisson, Binomial, are members of this family. UNM
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Generalized linear models (GLM) Independent random variables Y 1 , Y 2 , . . . , Y n where each Y i f ( y i | θ i ) = exp [ a ( y i ) b ( θ i ) + c ( θ i ) d ( y i ); i = 1 , 2 , . . . , n ] E ( Y i ) = μ i is a function of θ i . For a GLM, g ( μ i ) = x t i β where x t i = ( x i 1 , x i 2 , . . . , x ip ) set of covariates (predictors) for Y i . β t = ( β 1 , β 2 , . . . , β p ) set of regression coefficients. g ( · ) is the link function . Monotone and differentiable. g ( · ) is a modeling choice . Linear regression: g ( μ i ) = μ i . UNM
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Example 3.4 Y 1 , Y 2 , . . . , Y n are n-independent success-failure trials. P ( Y i = 1 ) = π and P ( Y i = 0 ) = 1 - π ., π is the probability of success. Probability function of Y i is f ( y i | π ) = π y i ( 1 - π ) 1 - y i ; y i = 0 , 1 E ( Y i ) = π but π is between 0 and 1.
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