Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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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