Details
| Original language | English |
|---|---|
| Pages (from-to) | 5997-6013 |
| Number of pages | 17 |
| Journal | Communications in Statistics - Theory and Methods |
| Volume | 50 |
| Issue number | 24 |
| Early online date | 13 Mar 2020 |
| Publication status | Published - 2021 |
Abstract
With any symmetric distribution μ on the real line we may associate a parametric family of noncentral distributions as the distributions of (Formula presented.) where X is a random variable with distribution μ. The classical case arises if μ is the standard normal distribution, leading to the noncentral chi-squared distributions. It is well known that these may be written as Poisson mixtures of the central chi-squared distributions with odd degrees of freedom. We obtain such mixture representations for the logistic distribution and for the hyperbolic secant distribution. We also derive alternative representations for chi-squared distributions and relate these to representations of the Poisson family. While such questions originated in parametric statistics they also appear in the context of the generalized second Ray–Knight theorem, which connects Gaussian processes and local times of Markov processes.
Keywords
- mixture distribution, Noncentral distribution, Poisson family, Primary 62E10, Ray–Knight theorem, Secondary 60E05
ASJC Scopus subject areas
- Mathematics(all)
- Statistics and Probability
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In: Communications in Statistics - Theory and Methods, Vol. 50, No. 24, 2021, p. 5997-6013.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Mixture representations of noncentral distributions
AU - Baringhaus, Ludwig
AU - Grübel, Rudolf
N1 - Publisher Copyright: © 2020 Taylor & Francis Group, LLC.
PY - 2021
Y1 - 2021
N2 - With any symmetric distribution μ on the real line we may associate a parametric family of noncentral distributions as the distributions of (Formula presented.) where X is a random variable with distribution μ. The classical case arises if μ is the standard normal distribution, leading to the noncentral chi-squared distributions. It is well known that these may be written as Poisson mixtures of the central chi-squared distributions with odd degrees of freedom. We obtain such mixture representations for the logistic distribution and for the hyperbolic secant distribution. We also derive alternative representations for chi-squared distributions and relate these to representations of the Poisson family. While such questions originated in parametric statistics they also appear in the context of the generalized second Ray–Knight theorem, which connects Gaussian processes and local times of Markov processes.
AB - With any symmetric distribution μ on the real line we may associate a parametric family of noncentral distributions as the distributions of (Formula presented.) where X is a random variable with distribution μ. The classical case arises if μ is the standard normal distribution, leading to the noncentral chi-squared distributions. It is well known that these may be written as Poisson mixtures of the central chi-squared distributions with odd degrees of freedom. We obtain such mixture representations for the logistic distribution and for the hyperbolic secant distribution. We also derive alternative representations for chi-squared distributions and relate these to representations of the Poisson family. While such questions originated in parametric statistics they also appear in the context of the generalized second Ray–Knight theorem, which connects Gaussian processes and local times of Markov processes.
KW - mixture distribution
KW - Noncentral distribution
KW - Poisson family
KW - Primary 62E10
KW - Ray–Knight theorem
KW - Secondary 60E05
UR - http://www.scopus.com/inward/record.url?scp=85081734326&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2206.10236
DO - 10.48550/arXiv.2206.10236
M3 - Article
AN - SCOPUS:85081734326
VL - 50
SP - 5997
EP - 6013
JO - Communications in Statistics - Theory and Methods
JF - Communications in Statistics - Theory and Methods
SN - 0361-0926
IS - 24
ER -