Assume {P, } is L1-differentiable, so that (14.93) holds. For simplicity, assume k = 1

Assume {Pθ, θ ∈ } is L1-differentiable, so that (14.93) holds.

For simplicity, assume k = 1 (but the problem generalizes). Let φ(·) be uniformly bounded and set β(θ) = Eθ[φ(X)]. Show β

(θ) exists at θ0 and

β

(θ0) =



φ(x)ζ(x, θ0)μ(dx) . (14.94)

Hence, if {Pθ} is q.m.d. at θ0 with derivative η(·, θ0), then

β

(θ0) =



φ(x)η˜(x, θ0)pθ0 (x)μ(dx) , (14.95)

where η˜(x, θ0) = 2η(x, θ0)/p 1/2

θ0 (x). More generally, if X1,..., Xn are i.i.d. Pθ and

φ(X1,..., Xn) is uniformly bounded, then β(θ) = Eθ[φ(X1,..., Xn)] is differentiable at θ0 with

β
(θ0) = 
··· 
φ(x1,..., xn)
n i=1 η˜(xi, θ0)
n i=1 pθ0 (xi)μ(dx1)··· μ(dxn) .
(14.96)
Section 14.3