
140 MARGINAL INFERENTIAL MODELS
where U
1
(α) and U
2
(α) are distributed as, respectively,
log
1
n
n
∑
i=1
U
0
i
α
and
1
n
n
∑
i=1
log
U
0
i
α
.
That the distributions of U
1
(α) and U
2
(α) depend on the nuisance parameter α
makes this a non-regular problem.
Next, define V
1
= U
1
(α) and V
2
= U
1
(α) −U
2
(α). For notational simplicity,
we have omitted the dependence of (V
1
,V
2
) on α. It is easy to check that nαe
V
1
has a gamma distribution with shape parameter nα; write F
nα
for the corresponding
gamma distribution function. Let κ(V
2
) be an estimator of α based on V
2
alone. This
estimator could be a moment estimator or perhaps a maximum likelihood estimator
based on the the marginal distrib ...