Fitting parametric sets to probability distributions

  • Meelis Käärik University of Tartu
  • Kalev Pärna University of Tartu
Keywords: approximation of distributions, loss-function, discrepancy function, consistency, M-estimation, k-centres


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.


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Author Biographies

Meelis Käärik, University of Tartu

Institute of Mathematical Statistics

Kalev Pärna, University of Tartu

Institute of Mathematical Statistics