Construction of Process Differentiable Representations for Parametric Families of Distributions

P. W. Glynn

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

Given a parametric family of distributions {F(θ, •): θ ∈(a,b)}, we consider the problem of constructing a process {X(θ): θ∈(a,b)}┬ásuch 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, •).