Appendix A
Mathematical Comments
A.1 RegularityConditions
These are the regularity conditions referred to in Sections 6.4 and 6.5 of the text.
A discussion of these conditions can be found in Chapter 6 of Lehmann and Casella
(1998).
LetXhave pdff(x;θ), whereθ∈Ω⊂Rp. For these assumptions,Xcan be
either a scalar random variable or a random vector inRk. As in Section 6.4, let
I(θ)=[Ijk]denotethep×pinformation matrix given by expression (6.4.4). Also,
we will denote the true parameterθbyθ 0.
Assumptions A.1.1.Additional regularity conditions for Sections 6.4 and 6.5.
(R6):There exists an open subsetΩ 0 ⊂Ωsuch thatθ 0 ∈Ω 0 and all third partial
derivatives off(x;θ)exist for allθ∈Ω 0.
(R7)The following equations are true (essentially, we can interchange expectation
and differentiation):
Eθ
[
∂
∂θj
logf(x;θ)
]
=0, forj=1,...,p
Ijk(θ)=Eθ
[
−
∂^2
∂θj∂θk
logf(x;θ)
]
, forj, k=1,...,p.
(R8)For al lθ∈Ω 0 ,I(θ)is positive definite.
(R9) There exist functionsMjkl(x)such that
∣
∣
∣
∣
∂^3
∂θj∂θkθl
logf(x;θ)
∣
∣
∣
∣≤Mjkl(x), for allθ∈Ω^0 ,
and
Eθ 0 [Mjkl]<∞, for allj, k, l∈ 1 ,...,p.
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