ST 762, HOMEWORK 1 EXTRA PROBLEMS, FALL 2007
These problems are from previous years and are for you to work on or not as you choose; they are
not to be turned in. You should be familiar with the concepts covered by these problems for the
midterm test. Solutions will be posted when the homework problems to be turned in are due.
1. Suppose we have independent pairs (Yj,xj), j= 1,...,n.
(a) Suppose that E(Yj|xj) = f(xj,β), j = 1,...,n, and that the conditional distribution of
Yjgiven xjis Poisson. Write down the corresponding loglikelihood (conditional on xj) for β
and derive the resulting estimating equation to be solved to obtain the maximum likelihood
estimator for β.
(b) Now suppose that E(Yj|xj) = kjf(xj,β), j = 1,...,n, and that Yjis the number of
successes in kjindependent trials, so that Yjgiven xjfollows a binomial distribution.
(c) What features do the estimating equations in (a) and (b) under these different distribu-
tional assumptions have in common?
2. Suppose we have independent pairs (Yj,xj), j= 1,...,n, such that
E(Yj|xj) = f(xj,β),var(Yj|xj) = σ2f2(xj,β), j = 1,...,n,
with σknown. For some of the calculations below, you may find it convenient to define
λβ(xj,β) = ∂/∂β log f(xj,β).
(a) Assuming that the distribution of Yjgiven xjis gamma, write down the corresponding
loglikelihood for β, and derive the form of the estimating equation for the maximum likelihood
estimator for β.
(b) Same as in (a), but now assuming the distribution of Yj|xjis normal.
(c) Same as in (a), but now assuming the distribution of Yj|xjis lognormal. (If Z∼N(m, γ2),
then Y= exp(Z) has a lognormal distribution.)
(d) Compare each estimating equation in (a) – (c) to the equations you found in Problem 1(a)
and (b), commenting specifically on similarities or differences in the general forms of these
equations. Do any of these equations share the common features you noted in 1(c)?
(e) Compare the estimating equations you found in (a) and (b) of this problem. State explic-
itly what features they share and what features are different. Give a possible reason for the
differences you observe.
3. More on transformations for nonlinear models. An assumption underlying the “Transform
Both Sides” (TBS) model discussed on pages 36–38 of the class notes is that a single trans-
formation can be used to achieve both constant variance and normality on the transformed
scale. For independent pairs (Yj,xj), j = 1,...,n, a generalization of this model has
E{h(Yj, λ)|xj}=h{f(xj,β), λ},var{h(Yj, λ)|xj}=σ2q2(xj, θ) (1)
for some transformation hdepending on a scalar parameter λ. Note that this model implies
that, after transformation, variance is not constant but depends on a function qof xjthat is
known if θis known.
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