Suppose {P, } is a model with an open subset of IRk , and having densities p(x) with respect to . Define the

Suppose {Pθ, θ ∈ } is a model with an open subset of IRk , and having densities pθ(x) with respect to μ. Define the model to be L1-differentiable at

θ0 if there exists a vector of real-valued functions ζ(·, θ0) such that



|pθ0+h(x) − pθ0 (x) − ζ(x, θ0), h|dμ(x) = o(|h|) (14.93)

as |h| → 0. Show that, if the family is q.m.d. at θ0 with q.m. derivative η(·, θ0), then it is L1-differentiable with

ζ(x, θ0) = 2η(x, θ0)p 1/2

θ0 (x) , but the converse is false.