Discrete mixture representations of parametric distribution families: Geometry and statistics

Ludwig Baringhaus; Rudolf Grübel

doi:10.48550/arXiv.2206.11094

Details

Original language	English
Pages (from-to)	37-70
Number of pages	34
Journal	Electronic journal of statistics
Volume	15
Issue number	1
Publication status	Published - 2021

Abstract

We investigate existence and properties of discrete mixture representations P_θ = ∑_i_∈Ew_θ(i) Q_i for a given family P_θ, θ ∈Θ, of probability measures. The noncentral chi-squared distributions provide a classi-cal example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher in-formation, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.

Keywords

Asymptotic efficiency, Barycenter convexity, Chi-squared distribution families, EM algorithm, Estimation of mixing probabilities, Fisher information, Mixture distribution

ASJC Scopus subject areas

Mathematics(all)
Statistics and Probability
Decision Sciences(all)
Statistics, Probability and Uncertainty

Cite this

Discrete mixture representations of parametric distribution families: Geometry and statistics. / Baringhaus, Ludwig; Grübel, Rudolf.
In: Electronic journal of statistics, Vol. 15, No. 1, 2021, p. 37-70.

Research output: Contribution to journal › Article › Research › peer review

Baringhaus, L & Grübel, R 2021, 'Discrete mixture representations of parametric distribution families: Geometry and statistics', Electronic journal of statistics, vol. 15, no. 1, pp. 37-70. https://doi.org/10.48550/arXiv.2206.11094, https://doi.org/10.1214/20-EJS1795

Baringhaus, L., & Grübel, R. (2021). Discrete mixture representations of parametric distribution families: Geometry and statistics. Electronic journal of statistics, 15(1), 37-70. https://doi.org/10.48550/arXiv.2206.11094, https://doi.org/10.1214/20-EJS1795

Baringhaus L, Grübel R. Discrete mixture representations of parametric distribution families: Geometry and statistics. Electronic journal of statistics. 2021;15(1):37-70. doi: 10.48550/arXiv.2206.11094, 10.1214/20-EJS1795

Baringhaus, Ludwig ; Grübel, Rudolf. / Discrete mixture representations of parametric distribution families : Geometry and statistics. In: Electronic journal of statistics. 2021 ; Vol. 15, No. 1. pp. 37-70.

Download

@article{bded1add09f344f8be229e9f52270cbe,

title = "Discrete mixture representations of parametric distribution families: Geometry and statistics",

abstract = "We investigate existence and properties of discrete mixture representations Pθ = ∑i∈Ewθ(i) Qi for a given family Pθ, θ ∈Θ, of probability measures. The noncentral chi-squared distributions provide a classi-cal example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher in-formation, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.",

keywords = "Asymptotic efficiency, Barycenter convexity, Chi-squared distribution families, EM algorithm, Estimation of mixing probabilities, Fisher information, Mixture distribution",

author = "Ludwig Baringhaus and Rudolf Gr{\"u}bel",

year = "2021",

doi = "10.48550/arXiv.2206.11094",

language = "English",

volume = "15",

pages = "37--70",

journal = "Electronic journal of statistics",

issn = "1935-7524",

publisher = "Institute of Mathematical Statistics",

number = "1",

}

Download

TY - JOUR

T1 - Discrete mixture representations of parametric distribution families

T2 - Geometry and statistics

AU - Baringhaus, Ludwig

AU - Grübel, Rudolf

PY - 2021

Y1 - 2021

N2 - We investigate existence and properties of discrete mixture representations Pθ = ∑i∈Ewθ(i) Qi for a given family Pθ, θ ∈Θ, of probability measures. The noncentral chi-squared distributions provide a classi-cal example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher in-formation, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.

AB - We investigate existence and properties of discrete mixture representations Pθ = ∑i∈Ewθ(i) Qi for a given family Pθ, θ ∈Θ, of probability measures. The noncentral chi-squared distributions provide a classi-cal example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher in-formation, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.

KW - Asymptotic efficiency

KW - Barycenter convexity

KW - Chi-squared distribution families

KW - EM algorithm

KW - Estimation of mixing probabilities

KW - Fisher information

KW - Mixture distribution

UR - http://www.scopus.com/inward/record.url?scp=85099845301&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2206.11094

DO - 10.48550/arXiv.2206.11094

M3 - Article

AN - SCOPUS:85099845301

VL - 15

SP - 37

EP - 70

JO - Electronic journal of statistics

JF - Electronic journal of statistics

SN - 1935-7524

IS - 1

ER -

Research@Leibniz University