Showing papers on "Natural exponential family published in 2017"
TL;DR: In this article, a new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility, namely one-parameter exponential distribution, for data analysis purposes.
Abstract: A new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility. A special case has been considered in detail, namely one-parameter exponential distribution. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, mode, moment-generating function, mean residual lifetime, stochastic orders, order statistics, and expression of the entropies, are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non linear equations only. Further, we consider an extension of the two-parameter exponential distribution also, mainly for data analysis purposes. Two datasets have been analyzed to show how the proposed models work in practice.
211 citations
TL;DR: The generalized transmuted-G (G-G) family as mentioned in this paper extends the G-G class with explicit expressions for the ordinary and incomplete moments, generating function, Renyi and Shannon entropies, order statistics and probability weighted moments.
Abstract: We introduce a new class of continuous distributions called the generalized transmuted-G family which extends the transmuted-G class. We provide six special models of the new family. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Renyi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The flexibility of the proposed family is illustrated by means of three applications to real data sets.
126 citations
TL;DR: The R package KFAS is described for state space modeling with the observations from an exponential family, namely Gaussian, Poisson, binomial, negative binomial and gamma distributions.
Abstract: State space modeling is an efficient and flexible method for statistical inference of a broad class of time series and other data. This paper describes the R package KFAS for state space modeling w ...
70 citations
TL;DR: In this article, a new family of distributions based on a one-parameter distribution exhibiting bathtub-shaped hazard rates is introduced, and the mathematical properties of the family and its parameters are studied by the method of maximum likelihood.
Abstract: We introduce a new family of distributions based on a one-parameter distribution exhibiting bathtub-shaped hazard rates. We study the mathematical properties of the family and estimate its parameters by the method of maximum likelihood. Finally, the usefulness of the family is illustrated using a real dataset.
60 citations
TL;DR: In this article, a new class of continuous distributions called the exponentiated Weibull-H family is proposed and studied, which is a family of probability distributions extended by Bourguignon et al.
Abstract: A new class of continuous distributions called the exponentiated Weibull-H family is proposed and studied. The proposed class extends the Weibull-H family of probability distributions introduced by Bourguignon et al. (J Data Sci 12:53–68, 2014). Some special models of the new family are presented. Its basic mathematical properties including explicit expressions for the ordinary and incomplete moments, quantile and generating function, Renyi and Shannon entropies, order statistics, and probability weighted moments are derived. The maximum-likelihood method is adopted to estimate the model parameters and a simulation study is performed. The flexibility of the generated family is proved empirically by means of two applications to real data sets.
49 citations
TL;DR: The potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data is investigated and various generalizations of the two-parameters exponential distribution are compared using maximum likelihood estimation.
Abstract: In this article, we investigate the potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data. We compare it with various generalizations of the two-parameter exponential distribution using maximum likelihood estimation. Some mathematical properties of the new extended model including expressions for the quantile and moments are investigated. We propose a location-scale regression model, based on the log-transmuted generalized exponential distribution. Two applications with real data are given to illustrate the proposed family of lifetime distributions.
38 citations
TL;DR: The problem of sequential estimation where information in the network diffuses with time is formulated abstractly and independently from any particular model to reach a generic solution that is applicable to a wide class of popular models and based on the exponential family of distributions.
Abstract: Diffusion networks where nodes collaboratively estimate the parameters of stochastic models from shared observations and other estimates have become an established research topic. In this paper the problem of sequential estimation where information in the network diffuses with time is formulated abstractly and independently from any particular model. The objective is to reach a generic solution that is applicable to a wide class of popular models and based on the exponential family of distributions. The adopted Bayesian and information-theoretic paradigms provide probabilistically consistent means for incorporation of shared observations in the implemented estimation of the unknowns by the nodes as well as for effective combination of the “knowledge” of the nodes over the network. It is shown and illustrated on four examples that under certain conditions, the resulting algorithms are analytically tractable, either directly or after simple approximations. The examples include linear regression, Kalman filtering, logistic regression, and inference of an inhomogeneous Poisson process. The first two examples have their more or less direct counterparts in the state-of-the-art diffusion literature whereas the latter two are new.
34 citations
TL;DR: In this article, a generalization of the length-biased exponential distribution called the Marshall-Olkin length biased exponential distribution (MLED) is proposed, and the parameters of the proposed model are estimated by maximum likelihood estimation method.
29 citations
TL;DR: In this paper, the authors established recurrence relations for the single and product moments of order statistics from the extended exponential distribution and tabulated the means, variances and covariances of all order statistics for all sample sizes.
Abstract: The extended exponential distribution due to Nadarajah and Haghighi (2011) is an alternative to and always provides better fits than the gamma, Weibull and the exponentiated exponential distributions whenever the data contains zero values. We establish recurrence relations for the single and product moments of order statistics from the extended exponential distribution. These recurrence relations enable computation of the means, variances and covariances of all order statistics for all sample sizes in a simple and efficient manner. By using these relations, we tabulate the means, variances and covariances of order statistics and derive best linear unbiased estimates of the extended exponential distribution. Finally, a data application is provided.
26 citations
Posted Content•
TL;DR: In this article, the authors derived analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modeled according to a mixture of exponential distributions.
Abstract: The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modeled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al, 2016), Gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.
19 citations
Posted Content•
TL;DR: In this paper, a conjugate multivariate distribution is proposed to analyze high-dimensional dependent data that are distributed according to a member from the natural exponential family of distributions. But it is not suitable for non-Gaussian data models, as is the case here.
Abstract: We introduce a Bayesian approach for analyzing (possibly) high-dimensional dependent data that are distributed according to a member from the natural exponential family of distributions. This problem requires extensive methodological advancements, as jointly modeling high-dimensional dependent data leads to the so-called "big n problem." The computational complexity of the "big n problem" is further exacerbated when allowing for non-Gaussian data models, as is the case here. Thus, we develop new computationally efficient distribution theory for this setting. In particular, we introduce the "conjugate multivariate distribution," which is motivated by the univariate distribution introduced in Diaconis and Ylvisaker (1979). Furthermore, we provide substantial theoretical and methodological development including: results regarding conditional distributions, an asymptotic relationship with the multivariate normal distribution, conjugate prior distributions, and full-conditional distributions for a Gibbs sampler. To demonstrate the wide-applicability of the proposed methodology, we provide two simulation studies and three applications based on an epidemiology dataset, a federal statistics dataset, and an environmental dataset, respectively.
TL;DR: In this paper, the authors present an overview of the mean value parametrization and characterization property of the variance function for NEF's and introduce the relationships existing between the NEF generating measure, Laplace transform and variance function.
Abstract: It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework The main aim of the paper is to expose the mean value parametrization for possible statistical applications The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF’s In particular it introduces the relationships existing between the NEF’s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation Some classes of polynomial variance functions are revisited for illustration The corresponding NEF’s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution
TL;DR: In this paper, the authors propose a way to construct fiducial distributions for a multidimensional parameter using a step-by-step conditional procedure related to the inferential importance of the components of the parameter.
TL;DR: In this paper, a new five-parameter continuous model called the beta generalized Gompertz distribution is introduced and studied, which consists of the GOMpertz, generalized GOMERTZ, beta GOMETZ, generalized exponential, beta generalized exponential and exponential distributions as special submodels.
25 Oct 2017
TL;DR: In this article, the differential calculus was used to obtain some classes of ordinary differential equations (ODE) for the probability density function, quantile function, survival function, inverse survival function and reversed hazard function of the exponentiated generalized exponential distribution.
Abstract: In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODE) for the probability density function, quantile function, survival function, inverse survival function, hazard function and reversed hazard function of the exponentiated generalized exponential distribution. The stated necessary conditions required for the existence of the ODEs are consistent with the various parameters that defined the distribution. Solutions of these ODEs by using numerous available methods are new ways of understanding the nature of the probability functions that characterize the distribution. The method can be extended to other probability distributions and can serve an alternative to approximation
TL;DR: In this paper, the authors introduce a method which allows perfect sampling from random graph models in exponential family form ("exponential family random graph" models), using a variant of Coupling From The Past.
Abstract: Generation of deviates from random graph models with non-trivial edge dependence is an increasingly important problem. Here, we introduce a method which allows perfect sampling from random graph models in exponential family form ("exponential family random graph" models), using a variant of Coupling From The Past. We illustrate the use of the method via an application to the Markov graphs, a family that has been the subject of considerable research. We also show how the method can be applied to a variant of the biased net models, which are not exponentially parameterized.
TL;DR: In this article, a flexible 3-parameter generalization of the exponential distribution is introduced based on the binomial exponential 2 (BE2) distribution, which exhibits decreasing, increasing and bathtub-shaped hazard rates, so it turns out to be quite flexible for analyzing nonnegative real life data.
Abstract: Developing statistical methods to model hydrologic events is always interesting for both statisticians and hydrologists, because of its importance in hydraulic structures design and water resource planning. Because of this, a flexible 3-parameter generalization of the exponential distribution is introduced based on the binomial exponential 2 (BE2) distribution [2]. The proposed distribution involving the exponential, gamma and BE2 distributions as submodels; and it exhibits decreasing, increasing and bathtub-shaped hazard rates, so it turns out to be quite flexible for analyzing non-negative real life data. Some statistical properties, parameters estimation and information matrix of the distribution are investigated. The proposed distribution, Gumbel, generalized Logistic and other distributions are utilized to model and fit two hydrologic data sets. The distribution is shown to be more appropriate to the data than the compared distributions using the selection criteria: average scaled absolute er...
TL;DR: The Modified Power Function (MPF) distribution as discussed by the authors is an extension of the one parameter Power function distribution, it enjoys a close form distributional expression and suggests the use of maximum likelihood method of parameter estimation for estimating the parameters of the new distribution.
Abstract: Recently, a lot of new, improved, flexible and robust probability distributions have been developed from the existing distributions to encourage their applications in diverse fields. This paper proposes a new lifetime distribution called the Modified Power function (MPF) distribution, the distribution belongs to the Marshall-Olkin-G family of distribution and it’s an extension of the one parameter Power function distribution. The MPF distribution enjoys a close form distributional expression. Some of its statistical properties including possible transformations are presented. The paper suggests the use of maximum likelihood method of parameter estimation for estimating the parameters of the new distribution. The applicability of the distribution was illustrated with two real data-sets and its goodness-of-fit was compared with that of the Exponential, Weibull, Lindley Exponential, Exponentiated Exponential, Kumaraswamy, Power function and Beta distributions by using the AIC, AICc, CAIC, BIC, HQC, W∗ and A∗...
TL;DR: In this article, the Marshall-Olkin generalized Erlang-truncated exponential (MOGETE) distribution was introduced as a generalization of the ETE distribution.
Abstract: This article introduces the Marshall–Olkin generalized Erlang-truncated exponential (MOGETE) distribution as a generalization of the Erlang-truncated exponential (ETE) distribution. The hazard rate of the new distribution could be increasing, decreasing or constant. Explicit-closed form mathematical expressions of some of the statistical and reliability properties of the new distribution were given and the method of maximum likelihood estimation was used to estimate the model parameters. The usefulness and flexibility of the new distribution was illustrated with two real and uncensored lifetime data-sets. The MOGETE distribution with a smaller goodness of fit statistics always emerged as a better candidate for the data-sets than the ETE, Exp Frechet and Exp Burr XII distributions.
27 Oct 2017
TL;DR: In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODE) for the probability density function, quantile function, survival function, inverse survival function and hazard function of the======Patrick Harris extended exponential distribution.
Abstract: In this paper, the differential calculus was used
to obtain some classes of ordinary differential equations (ODE)
for the probability density function, quantile function, survival
function, inverse survival function and hazard function of the
Harris extended exponential distribution. The case of reversed
hazard function was excluded because of its complexity. The
stated necessary conditions required for the existence of the
ODEs are consistent with the various parameters that defined
the distribution. Solutions of these ODEs by using numerous
available methods are new ways of understanding the nature of
the probability functions that characterize the distribution.
The method can be extended to other probability distributions,
functions and can serve an alternative to estimation and
approximation.
TL;DR: In this article, the main purpose of this paper is to present -generalized exponential distribution which among other things includes Generalized Exponential and Weibull Distributions as special cases.
Abstract: The main purpose of this paper is to present -Generalized Exponential Distribution which among other things includes Generalized Exponential and Weibull Distributions as special cases. Besides, we also obtain three-parameter extension of Generalized Exponential Distribution. We shall also discuss moment generating functions (MGFs) of these newly introduced distributions.
TL;DR: In this article, the generalized order statistics (GOS) from the extended exponential distribution (EE) was considered and exact explicit expressions as well as recurrence relations for the single, product, and conditional moments of generalized OS from the EE distribution were obtained.
Abstract: SYNOPTIC ABSTRACTThe extended exponential distribution due to Nadarajah and Haghighi (2011) is an alternative, and always provides better fits than the gamma, Weibull, and the generalized exponential distributions whenever the data contains zero values. In this article, we consider the generalized order statistics (GOS) from this distribution. We obtain exact explicit expressions as well as recurrence relations for the single, product, and conditional moments of generalized order statistics from the extended exponential (EE) distribution. Then, we use these results to compute the means, variances, and covariances of order statistics and record values for samples of different sizes for various values of the shape and scale parameters. Further, we compute the mean and variances for progressively Type II censored order statistics.
Journal Article•
TL;DR: In this article, Kumaraswamy Inverse Exponential distribution was applied to six real lifetime datasets and the performance was judged based on the log-likelihood and Akaike information criterion (AIC) values posed by the distributions.
Abstract: In this research, the Kumaraswamy Inverse Exponential distribution being a generalization of the Inverse Exponential distribution was applied to six real lifetime datasets. The idea is to assess its flexibility and superiority over its sub-models. Some other properties of the Kumaraswamy Inverse Exponential distribution were investigated in minute details. It was demonstrated and confirmed that the Kumaraswamy Inverse Exponential distribution performed better than the competing probability models except for data sets with variances far above the means. The performance was judged based on the log-likelihood and Akaike Information Criteria (AIC) values posed by the distributions.
TL;DR: In this article, the authors considered sequential point estimation of hazard rate function of an exponential distribution subject to the risk function given as, where 0 0 are known constants, and provided explicit formulas for the distribution of the total sample size, Nm, and of the expected value and risk of the estimator of the hazard rate of the exponential distribution.
Abstract: In this article, we consider sequential point estimation of hazard rate function of an exponential distribution () subject to the risk function given as , where 0 0 are known constants. We provide explicit formulas for the distribution of the total sample size, Nm, and of the expected value and risk of the estimator of the hazard rate function of the exponential distribution. In addition, we propose how to determine the parameter K( = K(m,A)) of the stopping variable Nm so that the risk is uniformly bounded by ω. In the end, the performances of the proposed methodology are investigated with the help of simulations and two applied examples.
TL;DR: Results based on the minimized log-likelihood, Akaike information criterion, Bayesian information criterion and the generalized Cramér–von Mises W⋆ statistics shows that the EETE distribution provides a more reasonable fit than the one based upon the other competing distributions.
TL;DR: In this paper, a numerical method to compute bivariate probability distributions from their Laplace transforms is presented, which consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF).
Abstract: A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties.
TL;DR: In this article, the authors proposed a distribution called Odds Exponential Log Logistic Distribution (OELLD), which is an odds family of distribution and hazard rate is an increasing and decreasing function based on the value of the parameter.
Abstract: We propose a distribution called Odds Exponential Log Logistic Distribution (OELLD), which is an odds family of distribution. Its hazard rate is an increasing and decreasing function based on the value of the parameter. Explicit expressions for the ordinary moments, L-moments, quantile, generating functions, Bonferroni Curve, Lorenz Curve, Gini's index and order statistics are derived. The parameters of the proposed distribution are estimated by using maximum likelihood method and also illustrated by a lifetime data set.
TL;DR: In this article, the authors studied the so-called Kumaraswamy Exponential Pareto (KEP) distribution and discussed the structural and mathematical properties of the KEP distribution.
Abstract: In this article we study the so-called Kumaraswamy Exponential Pareto (KEP) distribution. Several lifetime distributions such as the Kumaraswamy Weibull, Kumaraswamy exponential, Kumaraswamy Rayleigh, generalized Weibull, among several others are embedded in the proposed distribution. Various structural and mathematical properties of the KEP distribution are presented. We also discuss the parameter estimation and simulation methods. Real data set is used to illustrate the importance and flexibility of the proposed model.