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Natural Exponential Families with Quadratic Variance Functions
Annals of Statistics, 1982 The normal, Poisson, gamma, binomial, and negative binomial distributions are univariate natural exponential families with quadratic variance functions (the variance is at most a quadratic function of the mean). Only one other such family exists. Much theory is unified for these six natural exponential families by appeal to their quadratic variance ...
Carl N Morris
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Natural Exponential Families with Quadratic Variance Functions: Statistical Theory
Annals of Statistics, 1983 The normal, Poisson, gamma, binomial, negative binomial, and NEFGHS distributions are the six univariate natural exponential families (NEF) with quadratic variance functions (QVF). This sequel to Morris (1982) treats certain statistical topics that can be handled within this unified NEF-QVF formulation, including unbiased estimation, Bhattacharyya and ...
Carl N Morris
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Laplace Approximations for Natural Exponential Families with Cuts
Scandinavian Journal of Statistics, 1998 Standard and fully exponential form Laplace approximations to marginal densities are described and conditions under which these give exact answers are investigated. A general result is obtained and is subsequently applied in the case of natural exponential families with cuts, in order to derive the marginal posterior density of the mean parameter ...
M. Efstathiou +2 more
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Divergences Induced by the Cumulant and Partition Functions of Exponential Families and Their Deformations Induced by Comparative Convexity [PDF]
Entropy, 2023 Exponential families are statistical models which are the workhorses in statistics, information theory, and machine learning, among others. An exponential family can either be normalized subtractively by its cumulant or free energy function, or ...
Frank Nielsen
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Exponential Families with External Parameters
Entropy, 2022 In this paper we introduce a class of statistical models consisting of exponential families depending on additional parameters, called external parameters.
Marco Favretti
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J Multivar Anal, 2014 Diaconis and Ylvisaker (1979) give necessary conditions for conjugate priors for distributions from the natural exponential family to be proper as well as to have the property of linear posterior expectation of the mean parameter of the family. Their conditions for propriety and linear posterior expectation are also sufficient if the natural parameter ...
Hornik K, Grün B.
europepmc +4 more sources
Associated Natural Exponential Families and Elliptic Functions [PDF]
, 2016 This paper studies the variance functions of the natural exponential families (NEF) on the real line of the form \((Am^4+Bm^2+C)^{1/2}\) where m denoting the mean. Surprisingly enough, most of them are discrete families concentrated on \(\lambda \mathbb {Z}\) for some constant \(\lambda \) and the Laplace transform of their elements are expressed by ...
G. Letac
semanticscholar +3 more sources
On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families
Mathematics, 2021 Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ ...
Shaul K. Bar-Lev
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Small area estimation with spatially varying natural exponential families [PDF]
Journal of Statistical Computation and Simulation, 2015 Two-stage hierarchical models have been widely used in small area estimation to produce indirect estimates of areal means. When the areas are treated exchangeably and the model parameters are assumed to be the same over all areas, we might lose the ...
S. Sugasawa, Y. Kawakubo, Kota Ogasawara
semanticscholar +4 more sources
Extended Divergence on a Foliation by Continuous-Type Escort Distributions [PDF]
EntropyFrom an information geometric perspective, this study considers a natural foliation of dualistic structures associated with escort distributions of exponential families.
Keiko Uohashi
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