Bayesian estimation and prediction for certain type of mixtures

Our main, novel, result is that a certain non-central chi-squared mixture of noncentral chi-squared distributions is itself a scaled non-central chi-squared distribution.From this and a link to a known result on a mixture representation for a scaled central chi-squared distribution, numerous further mixture results, both old and new, ensue.These include mixture results for central F, non-central F and Libby-Novick distributions. The main result involves distributions all with the same degrees of freedom; it is also extended to the case where the mixing non-central chi-squared distribution has degrees of freedom an even number larger than that of the conditional noncentral chi-squared distribution, with further consequences pursued.

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“…This version of Corollary 1 was given by Perlman and Rasmussen (1975). See also L'Moudden and Marchand (2021, Example 3.2).…”

Section: Main Results and Consequences Thereofmentioning

confidence: 99%

A (non‐central) chi‐squared mixture of non‐central chi‐squareds is (non‐central) chi‐squared and related results, corollaries and applications

Jones

Marchand

2021

Stat

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“…Bayesian posterior analysis of MMN models relate to the general form X|K, θ ∼ f θ,K , K ∼ g , and θ ∼ π , (3.16) with observable X ∈ R d , density g of K free of θ, and π prior density for θ ∈ R d . Such a set-up leads to the following intermediate result, taken from [21].…”

Section: Posterior Densitiesmentioning

confidence: 99%

“…We make use of the following intermediate result provided in [21] and applicable to mixture models of the form:…”

Section: Predictive Densitiesmentioning

confidence: 99%

Bayesian inference and prediction for mean-mixtures of normal distributions

Bhagwat¹,

Marchand²

2022

Preprint

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We study frequentist risk properties of predictive density estimators for mean mixtures of multivariate normal distributions, involving an unknown location parameter θ ∈ R d , and which include multivariate skew normal distributions. We provide explicit representations for Bayesian posterior and predictive densities, including the benchmark minimum risk equivariant (MRE) density, which is minimax and generalized Bayes with respect to an improper uniform density for θ. For four dimensions or more, we obtain Bayesian densities that improve uniformly on the MRE density under Kullback-Leibler loss. We also provide plug-in type improvements, investigate implications for certain type of parametric restrictions on θ, and illustrate and comment the findings based on numerical evaluations.

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Bayesian inference and prediction for mean-mixtures of normal distributions

Bhagwat

Marchand

2023

Electron. J. Statist.

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Bayesian estimation and prediction for certain type of mixtures

Cited by 3 publications

References 24 publications

A (non‐central) chi‐squared mixture of non‐central chi‐squareds is (non‐central) chi‐squared and related results, corollaries and applications

A (non‐central) chi‐squared mixture of non‐central chi‐squareds is (non‐central) chi‐squared and related results, corollaries and applications

Bayesian inference and prediction for mean-mixtures of normal distributions

Bayesian inference and prediction for mean-mixtures of normal distributions

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