Showing papers on "Natural exponential family published in 2021"
TL;DR: The Weibull distribution is the most important distribution for problems in reliability as discussed by the authors, and it has been studied extensively in the literature, including in the context of the wider Weibbull-G family of distributions.
Abstract: The Weibull distribution is the most important distribution for problems in reliability. We study some mathematical properties of the new wider Weibull-G family of distributions. Some special models in the new family are discussed. The properties derived hold to any distribution in this family. We obtain general explicit expressions for the quantile function, ordinary and incomplete moments, generating function and order statistics. We discuss the estimation of the model parameters by maximum likelihood and illustrate the potentiality of the extended family with two applications to real data.
391 citations
TL;DR: In this paper, a new four-parameter lifetime model named the exponentiated generalized extended exponential distribution (EGEDE) was proposed, which includes the exponential and exponentiated exponential distributions, among others, and its hazard function can take the classic shapes: bath tub, inverted bathtub, increasing, decreasing and constant.
Abstract: We introduce and study a new four-parameter lifetime model named the exponentiated generalized extended exponential distribution. The proposed model has the advantage of including as special cases the exponential and exponentiated exponential distributions, among others, and its hazard function can take the classic shapes: bathtub, inverted bathtub, increasing, decreasing and constant, among others. We derive some mathematical properties of the new model such as a representation for the density function as a double mixture of Erlang densities, explicit expressions for the quantile function, ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating function, R´enyi entropy, density of order statistics and reliability. We use the maximum likelihood method to estimate the model parameters. Two applications to real data illustrate the flexibility of the proposed model.
21 citations
TL;DR: The Kummer gamma generalized gamma (KGGD) as discussed by the authors is an extension of the generalized gamma distribution with six-parameter Kummer beta generalized gamma, which can be expressed as a linear combination of generalized gamma densities.
Abstract: A new extension of the generalized gamma distribution with six- parameter called the Kummer beta generalized gamma distribution is introduced and studied. It contains at least 28 special models such as the beta generalized gamma, beta Weibull, beta exponential, generalized gamma, Weibull and gamma distributions and thus could be a better model for analyzing positive skewed data. The new density function can be expressed as a linear combination of generalized gamma densities. Various mathematical properties of the new distribution including explicit expressions for the ordinary and incomplete moments, generating function, mean deviations, entropy, density function of the order statistics and their moments are derived. The elements of the observed information matrix are provided. We discuss the method of maximum likelihood and a Bayesian approach to fit the model parameters. The superiority of the new model is illustrated by means of three real data sets.
10 citations
TL;DR: 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.
Abstract: 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.
7 citations
TL;DR: In this paper, a new family of models for lifetime data called generalized extended Weibull power series family of distributions was introduced by compounding generalized extended weibull distributions and power series distributions.
Abstract: In this study, we introduce a new family of models for lifetime data called generalized extended Weibull power series family of distributions by compounding generalized extended Weibull distributions and power series distributions. The compounding procedure follows the same setup carried out by Adamidis (1998). The proposed family contains all types of combinations between truncated discrete with generalized and non-generalized Weibull distributions. Some existing power series and subclasses of mixed lifetime distributions become special cases of the proposed family, such as the compound class of extended Weibull power series distributions proposed by Silva et al. (2013) and the generalized exponential power series distributions introduced by Mahmoudi and Jafari (2012). Some mathematical properties of the new class are studied, including the cumulative distribution function, density function, survival function, and hazard rate function. The method of maximum likelihood is used for obtaining a general setup for estimating the parameters of any distribution in this class. An expectation-maximization algorithm is introduced for estimating maximum likelihood estimates. Special subclasses and applications for some models in a real dataset are introduced to demonstrate the flexibility and the benefit of this new family.
6 citations
TL;DR: In this article, two extensions of the canonical Feller-Spitzer distribution from the class of Bessel densities are introduced, 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.
Abstract: We 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 Levy 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.
4 citations
TL;DR: In this article, two new classes of counting distributions based on natural exponential families on the real line were introduced, which are shown to be overdispersed and zero inflated in ascending order.
Abstract: In their fundamental paper on cubic variance functions (VFs), Letac and Mora (The Annals of Statistics , 1990) presented a systematic, rigorous and comprehensive study of natural exponential families (NEFs) on the real line, their characterization through their VFs and mean value parameterization. They presented a section that for some reason has been left unnoticed. This section deals with the construction of VFs associated with NEFs of counting distributions on the set of nonnegative integers and allows to find the corresponding generating measures. As EDMs are based on NEFs, we introduce in this paper two new classes of EDMs based on their results. For these classes, which are associated with simple VFs, we derive their mean value parameterization and their associated generating measures. We also prove that they have some desirable properties. Both classes are shown to be overdispersed and zero inflated in ascending order, making them as competitive statistical models for those in use in both, statistical and actuarial modeling. To our best knowledge, the classes of counting distributions we present in this paper, have not been introduced or discussed before in the literature. To show that our classes can serve as competitive statistical models for those in use (e.g. , Poisson, Negative binomial), we include a numerical example of real data. In this example, we compare the performance of our classes with relevant competitive models.
4 citations
TL;DR: In this paper, the authors introduced the beta linear failure rate geometric (BLFRG) distribution, which contains a number of distributions including the exponentiated linear failure rates geometric, exponential geometric, Rayleigh geometric, and Rayleigh and exponential distributions as special cases.
Abstract: This paper introduces the beta linear failure rate geometric (BLFRG) distribution, which contains a number of distributions including the exponentiated linear failure rate geometric, linear failure rate geometric, linear failure rate, exponential geometric, Rayleigh geometric, Rayleigh and exponential distributions as special cases. The model further generalizes the linear failure rate distribution. A comprehensive investigation of the model properties including moments, conditional moments, deviations, Lorenz and Bonferroni curves and entropy are presented. Estimates of model parameters are given. Real data examples are presented to illustrate the usefulness and applicability of the distribution.
3 citations
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TL;DR: In this paper, the authors presented results for bivariate exponential distributions which are represented by phase type distributions and showed how Markovian Arrival Processes (MAPs) with exponential marginal distribution can be generated from the phase type representations of exponential distributions and how the results for exponential distributions can be applied to define correlated hyperexponential or Erlang distributions.
Abstract: In this paper we present results for bivariate exponential distributions which are represented by phase type distributions. The paper extends results from previous publications [5, 14] on this topic by introducing new representations that require a smaller number of phases to reach some correlation coefficient and introduces different ways to describe correlation between exponentially distributed random variables. Furthermore, it is shown how Markovian Arrival Processes (MAPs) with exponential marginal distribution can be generated from the phase type representations of exponential distributions and how the results for exponential distributions can be applied to define correlated hyperexponential or Erlang distributions. As application examples we analyze two queueing models with correlated inter-arrival and service times.
1 citations
03 Jul 2021
TL;DR: In this paper, it was shown that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean, i.i.d.
Abstract: 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 .
TL;DR: In this article, the authors discuss several desirable properties of the inverse Gaussian (IG) family involving orthogonal polynomials and discuss the cumulant-generating function and associated properties.
Abstract: In this work, we discuss several desirable properties of the inverse Gaussian (IG) family involving orthogonal polynomials. In particular, we discuss the cumulant-generating function and associated...
TL;DR: The exponential and geometric distributions with the memoryless property are well-known continuous and discrete family of distributions as mentioned in this paper, respectively, and the memory-less property is emphasized in introd...
Abstract: The exponential and geometric distribution are well-known continuous and discrete family of distributions with the memoryless property, respectively. The memoryless property is emphasized in introd...