9.1 Let Y1,...,YN be independent random variables with Yi Po(i) and log i = 1 + J j=2 xi jj , i = 1,...,N.

9.1 Let Y1,...,YN be independent random variables with Yi ∼ Po(µi) and log µi = β1 +∑

J j=2 xi jβj

, i = 1,...,N.

a. Show that the score statistic for β1 is U1 = ∑

N i=1

(Yi − µi).

b. Hence, show that for maximum likelihood estimates µbi

, ∑µbi = ∑yi

.

c. Deduce that the expression for the deviance in (9.6) simplifies to (9.7)

in this case.