A moment-based test for the homogeneity in mixture natural exponential family with quadratic variance functions

2009 ◽  
Vol 79 (6) ◽  
pp. 828-834 ◽  
Author(s):  
Wei Ning ◽  
Sanguo Zhang ◽  
Chang Yu
Keyword(s):  
Exponential Family ◽  
Natural Exponential Family ◽  
Variance Functions

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 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.

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 Semiparametric Smoothing Spline in Joint Mean and Dispersion Models with Responses from the Biparametric Exponential Family: A Bayesian Perspective

2021 ◽  
Vol 9 (2) ◽  
pp. 351-367
Author(s):  
Héctor Zárate ◽  
Edilberto Cepeda
Keyword(s):  
Exponential Family ◽  
Parametric Models ◽  
Efficient Estimation ◽  
Smoothing Methods ◽  
Markov Chain Sampling ◽  
Variance Functions ◽  
Natural Connection ◽  
Mean And Variance ◽  
The Mean ◽  
Two Parameter

This article extends the fusion among various statistical methods to estimate the mean and variance functions in heteroscedastic semiparametric models when the response variable comes from a two-parameter exponential family distribution. We rely on the natural connection among smoothing methods that use basis functions with penalization, mixed models and a Bayesian Markov Chain sampling simulation methodology. The significance and implications of our strategy lies in its potential to contribute to a simple and unified computational methodology that takes into account the factors that affect the variability in the responses, which in turn is important for an efficient estimation and correct inference of mean parameters without the requirement of fully parametric models. An extensive simulation study investigates the performance of the estimates. Finally, an application using the Light Detection and Ranging technique, LIDAR, data highlights the merits of our approach.

scholarly journals ANOVA for Some Non-Normal Data by Inverting Reliability Estimators

2021 ◽  
Author(s):  
Robert Foster
Keyword(s):  
Analysis Of Variance ◽  
Generalized Linear Model ◽  
Exponential Family ◽  
Test Statistic ◽  
Natural Exponential Family ◽  
Standard Analysis ◽  
Conjugate Priors ◽  
Higher Power ◽  
Posterior Expectation ◽  
Mean Equality

Standard analysis of variance assumes observations are normally distributed within groups. This paper develops some analysis of variance tests for data which are Bernoulli, Poisson, exponential, or geometric distributed within groups. The tests are shown in Table 1. For natural exponential family data with conjugate priors for the distribution of means, reliability estimators directly estimate the posterior shrinkage. Using the linear posterior expectation induced by conjugate prior, a method is developed to construct an analysis of variance test by determining an appropriate transformation of a reliability estimator. The sampling distribution of the transformed reliability estimator under the assumption of group mean equality is derived to construct an appropriate test statistic. This method is used to invert the generalized KR21 estimators of Foster (2021) for some non-normal data, and it is also shown that the standard analysis of variance F-test statistic can be transformed into a consistent reliability estimator under the same assumptions. A limited simulation study shows that the inverted KR21 test has, in some scenarios, higher power than a standard analysis of variance or a generalized linear model analysis of variance.

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