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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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Increasing Hazard Rate of Mixtures for Natural Exponential Families [PDF]
Advances in Applied Probability, 2012 Hazard rates play an important role in various areas, e.g. reliability theory, survival analysis, biostatistics, queueing theory, and actuarial studies. Mixtures of distributions are also of great preeminence in such areas as most populations of components are indeed heterogeneous.
Shaul K. Bar‐Lev, Gérard Letac
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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.
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Natural Exponential Families with Quadratic Variance Functions [PDF]
The 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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Associated natural exponential families and elliptic functions [PDF]
, 2015 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 ...
Gérard Letac
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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 efficiency in the presence of spatial heterogeneity.
Shonosuke Sugasawa +2 more
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Natural Exponential Families with Quadratic Variance Functions: Statistical Theory [PDF]
The 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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Reproducibility and Natural Exponential Families with Power Variance Functions [PDF]
The Annals of Statistics, 1986 Let \(X_ 1\),..., \(X_ n\) be independent identically distributed random variables whose common distribution belongs to a family \({\mathcal F}=\{F_{\theta}\in \Theta \subset {\mathbb{R}}\}\) indexed by a parameter \(\theta\). We say that \({\mathcal F}\) is reproducible if there exists a sequence \(\{\) \(\alpha\) (n)\(\}\) such that \[ {\mathcal L ...
Shaul K. Bar‐Lev, Peter Enis
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Laplace Approximations for Natural Exponential Families with Cuts [PDF]
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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On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families [PDF]
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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