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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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Divergences Induced by the Cumulant and Partition Functions of Exponential Families and Their Deformations Induced by Comparative Convexity [PDF]
EntropyExponential 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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On standard conjugate families for natural exponential families with bounded natural parameter space
Journal of Multivariate Analysis, 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 ...
Kurt Hornik, Bettina Grün
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Location and scale behaviour of the quantiles of a natural exponential family [PDF]
ESAIM: Probability and Statistics, 2019 Let P0 be a probability on the real line generating a natural exponential family (Pt)t∈ℝ. Fix α in (0, 1). We show that the property that Pt((−∞, t)) ≤ α ≤ Pt((−∞, t]) for all t implies that there exists a number μα such that P0 is the Gaussian distribution N(μα, 1).
Mauro Piccioni +2 more
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Natural Real Exponential Families with Cubic Variance Functions [PDF]
The Annals of Statistics, 1990 Pursuing the classification initiated by Morris (1982), we describe all the natural exponential families on the real line such that the variance is a polynomial function of the mean with degree less than or equal to 3. We get twelve different types; the first six appear in the fundamental paper by Morris (1982); most of the other six appear as ...
Gérard Letac, Marianne Mora
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A construction of the UMVU estimator for simple quadratic natural exponential families
Journal of Multivariate Analysis, 2003 AbstractThis paper presents a construction of the uniformly minimum variance unbiased (UMVU) estimator of real-valued functions for the simple quadratic natural exponential families on Rd. A polynomial expansion of the estimator is derived and a condition for its existence is given. The exact variance of the UMVU estimator is calculated.
Denys Pommeret
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Cumulative distribution functions for the five simplest natural exponential families [PDF]
Metrika, 2019 Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\E(z^{X_a})=f(az)/f(a)$. Assume that $\Pr(X_a\leq k)$ has the form $c_k\int_a ^{\infty}u^k (du)$ for some number $c_k$ and some positive measure $ ,$ both independent of $a.$ We show that this asumption implies that
Gérard Letac
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Mathematics, 2023 The paper comprehensively studies the natural exponential family and its associated exponential dispersion model generated by the Landau distribution. These families exhibit probabilistic and statistical properties and are suitable for modeling skewed ...
Shaul K. Bar-Lev
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Cumulant-Based Goodness-of-Fit Tests for the Tweedie, Bar-Lev and Enis Class of Distributions
Mathematics, 2023 The class of natural exponential families (NEFs) of distributions having power variance functions (NEF-PVFs) is huge (uncountable), with enormous applications in various fields.
Shaul K. Bar-Lev +4 more
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Risks, 2022 The Lee–Carter model, the dominant mortality projection modeling in the literature, was criticized for its homoscedastic error assumption. This was corrected in extensions to the model based on the assumption that the number of deaths follows Poisson or ...
Yaser Awad, Shaul K. Bar-Lev, Udi Makov
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