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Approximation by mixtures of multivariate Erlang distributions

Hien Duy Nguyen

Abstract

We prove that finite multivariate Erlang mixture densities with a common rate parameter are dense in the class of probability densities on $\mathbb{R}_{+}^{d}$ that belong to $L^{p}$, for every dimension $d\in\mathbb{N}$ and every $1\le p<\infty$. The argument is constructive: the one-dimensional Szász--Mirakjan--Kantorovich operator yields Erlang mixture approximations, and its tensor product yields multivariate approximants with a common scale. We then obtain several quantitative consequences. These include compact-set uniform approximation bounds and, under local Hölder conditions of order $α\in(0,1]$, rates of order $n^{-α/2}$ as the common scale $1/n$ tends to zero, whole-domain convergence in weighted sup norms, weighted and unweighted $L^{p}$ rates, and explicit rates for finite mixtures indexed by the number of mixture components. In particular, if the approximating density is required to have at most $K$ mixture components, then on fixed compact cubes we obtain an algebraic rate of order $K^{-α/(2d)}$; in global weighted sup norms we obtain the explicit algebraic component-count rate $K^{-α/[2d(2d+α)]}$; and for $1<p<\infty$ we obtain corresponding weighted $L^{p}$ component-count rates. The results strengthen the weak-approximation theory for multivariate Erlang mixture distributions and yield immediate corollaries for broader classes such as product-gamma mixtures. \noindent\textbf{Keywords:} multivariate Erlang mixtures; Erlang distributions; Szász--Mirakjan--Kantorovich operator; density approximation; weighted $L^{p}$ approximation; approximation rates.

Approximation by mixtures of multivariate Erlang distributions

Abstract

We prove that finite multivariate Erlang mixture densities with a common rate parameter are dense in the class of probability densities on that belong to , for every dimension and every . The argument is constructive: the one-dimensional Szász--Mirakjan--Kantorovich operator yields Erlang mixture approximations, and its tensor product yields multivariate approximants with a common scale. We then obtain several quantitative consequences. These include compact-set uniform approximation bounds and, under local Hölder conditions of order , rates of order as the common scale tends to zero, whole-domain convergence in weighted sup norms, weighted and unweighted rates, and explicit rates for finite mixtures indexed by the number of mixture components. In particular, if the approximating density is required to have at most mixture components, then on fixed compact cubes we obtain an algebraic rate of order ; in global weighted sup norms we obtain the explicit algebraic component-count rate ; and for we obtain corresponding weighted component-count rates. The results strengthen the weak-approximation theory for multivariate Erlang mixture distributions and yield immediate corollaries for broader classes such as product-gamma mixtures. \noindent\textbf{Keywords:} multivariate Erlang mixtures; Erlang distributions; Szász--Mirakjan--Kantorovich operator; density approximation; weighted approximation; approximation rates.
Paper Structure (9 sections, 30 theorems, 132 equations)

This paper contains 9 sections, 30 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Notation and technical preliminaries
  3. Qualitative approximation theorems
  4. Convergence rates in scale
  5. Convergence rates in the number of components
  6. Proofs and technical results
  7. One-dimensional input, tensorization, and qualitative approximation
  8. Probabilistic preliminaries and scale-rate proofs
  9. Proofs for component-count approximation

Key Result

Theorem 2.3

Let $1\le p<\infty$. For every $n\in\mathbb{N}$, the operator $K_{n}$ maps $L^{p}(\mathbb{R}_{+})$ into itself, is a positive linear contraction, and satisfies $\left\lVert K_{n}g\right\rVert _{L^{p}(\mathbb{R}_{+})}\le\left\lVert g\right\rVert _{L^{p}(\mathbb{R}_{+})}$ for every $g\in L^{p}(\mathbb

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: One-dimensional $L^p$ convergence; Theorem 3.5 of AltomareCappellettiLeonessa2013
  • Proposition 2.4: One-dimensional Erlang mixture representation
  • Proposition 2.5: A probabilistic representation
  • Lemma 2.6: Moments of the displacement
  • Lemma 2.7: Weighted moments of $Y_{n,x}$
  • Remark 2.8
  • Theorem 3.1: Tensorized $L^p$ convergence
  • Proposition 3.2: Multivariate Erlang mixture formula
  • ...and 56 more