Construction of Process Differentiable Representations for Parametric Families of Distributions

P. W. Glynn

Technical Report, Mathematics Research Center, University of Wisconsin, Madison (1987)

• G87d.pdf

Given a parametric family of distributions {F(θ, •): θ ∈(a,b)}, we consider the problem of constructing a process {X(θ): θ∈(a,b)} such that: i.) for all θ∈(a,b), X(θ) has distribution F(θ, •), ii.) X(•) is differentiable in an L¹ sense at θ₀. This problem is motivated by certain computational questions associated with Monte Carlo estimation of the derivative of an expectation which depends on a parameter. A by-product of this work is a partial solution to the problem of constructing, for a given parametric family {F(θ, •)}, a function u such that u(θ, X(θ₀)) has distribution F(θ, •).