scholarly journals Reduction functions for the variance function of one-parameter natural exponential family

2018 ◽  
Vol 137 ◽  
pp. 319-325
Author(s):  
Xiongzhi Chen
Keyword(s):  
Exponential Family ◽  
Variance Function ◽  
Natural Exponential Family

scholarly journals On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1568
Author(s):  
Shaul K. Bar-Lev
Keyword(s):  
Power Function ◽  
Exponential Family ◽  
Probability Distributions ◽  
Infinite Divisibility ◽  
Variance Function ◽  
Exponential Families ◽  
Natural Exponential Family ◽  
Natural Exponential Families ◽  
Convolution Properties ◽  
The Relationship

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) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

scholarly journals Lévy processes time-changed by the first-exit time of the inverse Gaussian subordinator

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2545-2552
Author(s):  
Farouk Mselmi
Keyword(s):  
Lévy Processes ◽  
Exponential Family ◽  
Exit Time ◽  
Levy Processes ◽  
Variance Function ◽  
Inverse Gaussian ◽  
Natural Exponential Family ◽  
First Exit Time

This paper deals with a characterization of the first-exit time of the inverse Gaussian subordinator in terms of natural exponential family. This leads us to characterize, by means its variance function, the class of L?vy processes time-changed by the first-exit time of the inverse Gaussian subordinator.

scholarly journals KR20 and KR21 for Some Nondichotomous Data (It’s Not Just Cronbach’s Alpha)

2021 ◽  
pp. 001316442199253
Author(s):  
Robert C. Foster
Keyword(s):  
Exponential Family ◽  
Exponential Families ◽  
Natural Exponential Family ◽  
Cronbach’S Alpha ◽  
Natural Exponential Families ◽  
Cronbach's Alpha ◽  
Generalized Framework ◽  
The Common ◽  
Poisson Data ◽  
Poisson Count

This article presents some equivalent forms of the common Kuder–Richardson Formula 21 and 20 estimators for nondichotomous data belonging to certain other exponential families, such as Poisson count data, exponential data, or geometric counts of trials until failure. Using the generalized framework of Foster (2020), an equation for the reliability for a subset of the natural exponential family have quadratic variance function is derived for known population parameters, and both formulas are shown to be different plug-in estimators of this quantity. The equivalent Kuder–Richardson Formulas 20 and 21 are given for six different natural exponential families, and these match earlier derivations in the case of binomial and Poisson data. Simulations show performance exceeding that of Cronbach’s alpha in terms of root mean square error when the formula matching the correct exponential family is used, and a discussion of Jensen’s inequality suggests explanations for peculiarities of the bias and standard error of the simulations across the different exponential families.

scholarly journals On two extensions of the canonical Feller–Spitzer distribution

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Vladimir Vladimirovich Vinogradov ◽  
Richard Bruce Paris
Keyword(s):  
Hypergeometric Functions ◽  
Exponential Family ◽  
Special Functions ◽  
Natural Exponential Family ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Absolutely Continuous ◽  
The Mean ◽  
Monotone Densities ◽  
Mean Variance

AbstractWe introduce two extensions of the canonical Feller–Spitzer distribution from the class of Bessel densities, which comprise two distinct stochastically decreasing one-parameter families of positive absolutely continuous infinitely divisible distributions with monotone densities, whose upper tails exhibit a power decay. The densities of the members of the first class are expressed in terms of the modified Bessel function of the first kind, whereas the members of the second class have the densities of their Lévy measure given by virtue of the same function. The Laplace transforms for both these families possess closed–form representations in terms of specific hypergeometric functions. We obtain the explicit expressions by virtue of the particular parameter value for the moments of the distributions considered and establish the monotonicity of the mean, variance, skewness and excess kurtosis within the families. We derive numerous properties of members of these classes by employing both new and previously known properties of the special functions involved and determine the variance function for the natural exponential family generated by a member of the second class.

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