STA 7249: Generalized Linear Models
Spring 2004: Assignment 2
More problems will be added to this assignment. These are given now
so that you can get started on them.
1. (Exercise 5.10, McCulloch and Searle, 2001) Suppose that y1, . . . , ynare independent
with µi=E(yi) satisfying
log µi=xiβ(xiunivariate)
and
Var(yi) = φµi.
(a) Give the quasi-likelihood estimating equation for βand find the asymptotic vari-
ance of ˜
β, the “maximum quasi-likelihood estimator” (MQLE) of β.
(b) Suppose that y1, . . . , ynare normally distributed. Derive the likelihood equations
for βand φand the asymptotic variance of ˆ
β, the MLE of β.
(c) Calculate the ratio of the asymptotic variance of ˜
βto that of ˆ
β. For concreteness,
assume that n/2 of the observations have xi= 5 and the remaining n/2 have
xi= 10. Do the calculations for βequal to 0.1, 1, and 10.
2. (Exercise 4.14, McCullagh and Nelder, 1989) Suppose that Y∼Binm, eλ/(1 + eλ).
Show that Y0=m−Yis also binomially distributed, and that the induced parameter
is λ0=−λ. Consider
˜
λ= logY+c1
m−Y+c2.
as an estimator of λ. Show that in order that the estimator be equivariant under the
transformation Y7→ m−Y, we must have c1=c2.
3. (Exercise 4.15, McCullagh and Nelder, 1989) Suppose that Y∼Bin(m, π), 0 < π < 1
Show that for c > 0,
E{log(Y+c)}= log(mπ) + c
mπ −1−π
2mπ +O(m−3/2).(∗)
Find the corresponding expansion for log(m−Y+c), and use these results to deduce
that
E(˜
λ) = λ+(1 −2π)(c−1/2)
mπ(1 −π)+O(m−3/2).
This shows that choosing c= 1/2 makes ˜
λapproximately unbiased.
Hint: Write Y=mπ +pmπ(1 −π)Z, where Z= (Y−mπ)/pmπ(1 −π), and note
that Z=Op(1) as m→ ∞, since Zd
−→ N(0,1). Now use Taylor expansion to
approximate log{(Y+c)/(mπ)}. To make this completely rigorous, you will have to
explicitly show that the expectation of the error term is O(m−3/2). This takes some
work but can be done using Hoeffing’s inequality, which can be found, e.g., on page 75
of Serfling (1980), Approximation Theorems of Mathematical Statistics.
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