Fitting parametric sets to probability distributions

Abstract

We consider the problem of approximation of distributions by sets. Given a probability P on a metric space (S,d) and a class A of subsets of S, find an approximative set A∈ A that minimizes the mean discrepancy of a random point X ∼ P from the set A: 

W(A,P) = ∫_Sφ(d(x,A))P(dx) → min _{A∈ A}.

We are especially interested in the case of parametric approximative sets, A = {A(Θ) : Θ∈ T }, where the aim is to find a value of the parameter Θ∈ T which minimizes 

 W(Θ,P) = W(A(Θ,P))→ min _{Θ∈ T} .

Current article is an extension of Käärik and Pärna (2003) to the case where approximating sets A(Θ) are unions of k parametric sets of the same type. We will prove the convergence of optimal values of loss functions and the existence of optimal sets in some functional spaces.