Showing papers on "Natural exponential family published in 2013"
TL;DR: A new three-parameter exponential-type family of distributions which can be used in modeling survival data, reliability problems and fatigue life studies is introduced and maximum likelihood estimation of the unknown parameters of the new model for complete sample as well as for censored sample is discussed.
110 citations
TL;DR: The beta exponentiated Weibull distribution is introduced which extends recent models by Lee et al. and it is demonstrated that the density of the new distribution can be expressed as a linear combination of WeIBull densities.
Abstract: The Weibull distribution is one of the most important distributions in reliability. For the first time, we introduce the beta exponentiated Weibull distribution which extends recent models by Lee et al. [Beta-Weibull distribution: some properties and applications to censored data, J. Mod. Appl. Statist. Meth. 6 (2007), pp. 173–186] and Barreto-Souza et al. [The beta generalized exponential distribution, J. Statist. Comput. Simul. 80 (2010), pp. 159–172]. The new distribution is an important competitive model to the Weibull, exponentiated exponential, exponentiated Weibull, beta exponential and beta Weibull distributions since it contains all these models as special cases. We demonstrate that the density of the new distribution can be expressed as a linear combination of Weibull densities. We provide the moments and two closed-form expressions for the moment-generating function. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The density of...
87 citations
TL;DR: In this article, a discrete analogue of the generalized exponential distribution of Gupta and Kundu is introduced, which can be viewed as another generalization of the geometric distribution, i.e., the DGE2(α, p) distribution.
Abstract: In this paper, we shall attempt to introduce another discrete analogue of the generalized exponential distribution of Gupta and Kundu [Generalized exponential distributions, Aust. N. Z. J. Stat. 41(2) (1999), pp. 173–188], different to that of Nekoukhou et al. [A discrete analogue of the generalized exponential distribution, Comm. Stat. Theory Methods, to appear (2011)]. This new discrete distribution, which we shall call a discrete generalized exponential distribution of the second type (DGE2(α, p)), can be viewed as another generalization of the geometric distribution. We shall first study some basic distributional and moment properties, as well as order statistics distributions of this family of new distributions. Certain compounded DGE2(α, p) distributions are also discussed as the results of which some previous lifetime distributions such as that of Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42] follow as corollaries. Then, we w...
73 citations
TL;DR: In this paper, a general method for obtaining more flexible new distributions by compounding the extended Weibull and power series distributions was introduced, and the compounding procedure follows the same set-up carried out by Adamidis and Loukas (1998) and defines 68 new submodels.
73 citations
Journal Article•
TL;DR: In this paper, a new class of distributions called New Generalized Lindley Distribution (NGLD) is presented, which contains several distributions such as gamma, exponential and Lindley as special cases.
Abstract: In this paper, we present a new class of distributions called New Generalized Lindley Distribution(NGLD). This class of distributions contains several distributions such as gamma, exponential and Lindley as special cases. The hazard function, reverse hazard function, moments and moment generating function and inequality measures are are obtained. Moreover, we discuss the maximum likelihood estimation of this distribution. The usefulness of the new model is illustrated by means of two real data sets. We hope that the new distribution proposed here will serve as an alternative model to other models available in the literature for modelling positive real data in many areas.
58 citations
01 Feb 2013
TL;DR: In this paper, the authors proposed a new distribution called the beta generalized Rayleigh distribution, which contains as special sub-models some well-known distributions, such as the generalized R-Rayleigh distribution.
Abstract: For the first time, we propose a new distribution so-called the beta generalized Rayleigh distribution that contains as special sub-models some well-known distributions. Expansions for the cumulative distribution and density functions are derived. We obtain explicit expressions for the moments, moment generating function, mean deviations, Bonferroni and Lorenz curves and densities of the order statistics and their moments. We estimate the parameters by maximum likelihood and provide the observed information matrix. The usefulness of the new distribution is illustrated through two real data sets that show that it is quite flexible in analyzing positive data instead of the generalized Rayleigh and Rayleigh distributions.
55 citations
TL;DR: In this article, the complementary exponential power series distributions, with failure rate either increasing or decreasing, were introduced, where the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks.
Abstract: In this paper, we introduce the complementary exponential power series distributions, with failure rate either increasing, which is complementary to the exponential power series model proposed by Chahkandi & Ganjali (2009). The new class of distribution arises on a latent complementary risks scenarios, where the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks. This new class contains several distributions as particular case. The properties of the proposed distribution class are discussed such as quantiles, moments and order statistics. Estimation is carried out via maximum likelihood. Simulation results on maximum likelihood estimation are presented. An real application illustrate the usefulness of the new distribution class.
51 citations
TL;DR: In this article, the exact distributions of the MLEs of a two-parameter exponential distribution when the data are Type-I progressively hybrid censored are derived. But their results are not applicable to the case of hybrid censored data.
49 citations
TL;DR: In this paper, the authors generalize the exponential family of probability distributions by replacing the exponential function with a φ-function, resulting in a Ά-family of distributions, where the analogue of the cumulant generating function is a normalizing function.
Abstract: We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function is a normalizing function. We define the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.
45 citations
TL;DR: For the first time, a new five-parameter distribution, called the beta generalized gamma distribution, is introduced and studied, which contains at least 25 special sub-models such as the beta gamma, beta Weibull, beta exponential, generalized gamma (GG), WeIBull and gamma distributions and thus could be a better model for analysing positive skewed data.
Abstract: For the first time, a new five-parameter distribution, called the beta generalized gamma distribution, is introduced and studied. It contains at least 25 special sub-models such as the beta gamma, beta Weibull, beta exponential, generalized gamma (GG), Weibull and gamma distributions and thus could be a better model for analysing positive skewed data. The new density function can be expressed as a linear combination of GG densities. We derive explicit expressions for moments, generating function and other statistical measures. The elements of the expected information matrix are provided. The usefulness of the new model is illustrated by means of a real data set.
42 citations
TL;DR: In this paper, a new lifetime distribution by compounding exponential and Poisson-Lindley distributions, named the exponential Poisson−Lindley (EPL) distribution, was introduced.
Abstract: In this paper, we introduce a new lifetime distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. A practical situation where the EPL distribution is most appropriate for modelling lifetime data than exponential–geometric, exponential–Poisson and exponential–logarithmic distributions is presented. We obtain the density and failure rate of the EPL distribution and properties such as mean lifetime, moments, order statistics and Renyi entropy. Furthermore, estimation by maximum likelihood and inference for large samples are discussed. The paper is motivated by two applications to real data sets and we hope that this model will be able to attract wider applicability in survival and reliability.
TL;DR: The exponentiated generalized linear exponential distribution (EGLEDE) as discussed by the authors is a generalization of the LDE which generalizes the generalized linear failure rate (GLR) distribution.
TL;DR: A new three-parameter distribution motivated mainly by lifetime issues is introduced and two real data applications are described to show superior performance versus some known lifetime models.
Abstract: Many if not most lifetime distributions are motivated only by mathematical interest. Here, a new three-parameter distribution motivated mainly by lifetime issues is introduced. Some properties of the new distribution including estimation procedures, univariate generalizations and bivariate generalizations are derived. Two real data applications are described to show superior performance versus some known lifetime models.
TL;DR: In this paper, the density of the beta Weibull distribution is expressed as a mixture of Weibbull densities and two closed-form expressions are derived for their moments.
Abstract: In this paper, we study some mathematical properties of the beta Weibull (BW) distribution, which is a quite flexible model in analysing positive data. It contains the Weibull, exponentiated exponential, exponentiated Weibull and beta exponential distributions as special sub-models. We demonstrate that the BW density can be expressed as a mixture of Weibull densities. We provide their moments and two closed-form expressions for their moment-generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and two entropies. The density of the BW-order statistics is a mixture of Weibull densities and two closed-form expressions are derived for their moments. The estimation of the parameters is approached by two methods: moments and maximum likelihood. We compare the performances of the estimates obtained from both the methods by simulation. The expected information matrix is derived. For th...
TL;DR: Characteristics of standard conjugate priors and their induced posteriors in Bayesian inference for von Mises–Fisher distributions are discussed, using either the canonical natural exponential family or the more commonly employed polar coordinate parameterizations.
TL;DR: In this paper, a new family of distributions by adding a parameter to the Marshall-Olkin family was introduced, which is a generalization of the exponential distribution, and the shape properties, moments, distributions of the order statistics, entropies and estimation procedures are derived.
Abstract: We introduce a new family of distributions by adding a parameter to the Marshall–Olkin family of distributions Some properties of the new family of distributions are derived A particular case of the family, a three-parameter generalization of the exponential distribution, is given special attention The shape properties, moments, distributions of the order statistics, entropies and estimation procedures are derived An application to a real data set is discussed
Journal Article•
TL;DR: In this paper, the authors introduce a new distribution that is dependent on the exponential and Pareto distribution and present some properties such that the moment generated function, such as the mean, mode, median, variance, the r-th moment about the mean and about the origin, reliability, hazard functions, coefficients of variation, of sekeness and of kurtosis, can be estimated.
Abstract: In this paper we introduce a new distribution that is dependent on the Exponential and Pareto distribution and present some properties such that the moment generated function, mean, mode, median , variance , the r-th moment about the mean, the r-th moment about the origin, reliability , hazard functions , coefficients of variation ,of sekeness and of kurtosis. Finally, we estimate the parameter . Keyword: Exponential distribution, Pearson distribution, moment estimation
TL;DR: The complementary exponentiated exponential geometric distribution (CEG) as mentioned in this paper is a new family of lifetime distributions, which arises on a latent competing risk scenario, where the lifetime associated with a particular risk is not observable but only the maximum lifetime value among all risks.
Abstract: We proposed a new family of lifetime distributions, namely, complementary exponentiated exponential geometric distribution. This new family arises on a latent competing risk scenario, where the lifetime associated with a particular risk is not observable but only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments, rth moment of the ith order statistic, mean residual lifetime, and modal value. Inference is implemented via a straightforwardly maximum likelihood procedure. The practical importance of the new distribution was demonstrated in three applications where our distribution outperforms several former lifetime distributions, such as the exponential, the exponential-geometric, the Weibull, the modified Weibull, and the generalized exponential-Poisson distribution.
TL;DR: Gupta and Kundu as mentioned in this paper introduced a new class of weighted Marshall-Olkin bivariate exponential distributions, which has univariate WE marginals and can be used quite effectively to model lifetime data.
Abstract: Recently, Gupta and Kundu [R.D. Gupta and D. Kundu, A new class of weighted exponential distributions, Statistics 43 (2009), pp. 621–634] have introduced a new class of weighted exponential (WE) distributions, and this can be used quite effectively to model lifetime data. In this paper, we introduce a new class of weighted Marshall–Olkin bivariate exponential distributions. This new singular distribution has univariate WE marginals. We study different properties of the proposed model. There are four parameters in this model and the maximum-likelihood estimators (MLEs) of the unknown parameters cannot be obtained in explicit forms. We need to solve a four-dimensional optimization problem to compute the MLEs. One data set has been analysed for illustrative purposes and finally we propose some generalization of the proposed model.
TL;DR: In this paper, a spectrum of related characterizations of the exponential distribution are identified and verified, motivated by the observation that for a sample of size two from an exponential distribution, the largest order statistic is distributed as a convolution of two independent exponential random variables with distributions differing only in their intensity or rate parameter.
TL;DR: In this article, the authors introduced the beta log-normal (LN) distribution for which the LN distribution is a special case and derived expressions for its moments and for the moments of order statistics.
Abstract: For the first time, we introduce the beta log-normal (LN) distribution for which the LN distribution is a special case. Various properties of the new distribution are discussed. Expansions for the cumulative distribution and density functions that do not involve complicated functions are derived. We obtain expressions for its moments and for the moments of order statistics. The estimation of parameters is approached by the method of maximum likelihood, and the expected information matrix is derived. The new model is quite flexible in analysing positive data as an important alternative to the gamma, Weibull, generalized exponential, beta exponential, and Birnbaum–Saunders distributions. The flexibility of the new distribution is illustrated in an application to a real data set.
TL;DR: In this article, the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modeling has been investigated and necessary and sufficient conditions for a non-stationary subdivision to have the reproduction property of exponential polynomials.
Abstract: An important capability for a subdivision scheme is the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modeling. In this regards, this study first provides necessary and sufficient conditions for a non-stationary subdivision to have the reproducing property of exponential polynomials. Then, the approximation order of such non-stationary schemes is discussed to quantify their approximation power. Based on these results, we see that the exponential B-spline generates exponential polynomials in the associated spaces, but it may not reproduce any exponential polynomials. Thus, we present normalized exponential B-splines that reproduce certain sets of exponential polynomials. One interesting feature is that the set of exponential polynomials to be reproduced is varied depending on the normalization factor. This provides us with the necessary accuracy and flexibility in designing target curves and surfaces. Some numerical results are presented to support the advantages of the normalized scheme by comparing them to the results without normalization.
Proceedings Article•
01 Jan 2013
TL;DR: In this paper, the authors showed that a Bayesian prediction strategy with Jereys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if the optimal strategy can be calculated without knowing the time horizon in advance.
Abstract: We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012b) showed that a Bayesian prediction strategy with Jereys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if and only if the optimal strategy can be calculated without knowing the time horizon in advance. They put forward the question what families have exchangeable SNML strategies. This paper fully answers this open problem for onedimensional exponential families. The exchangeability can happen only for three classes of natural exponential family distributions, namely the Gaussian, Gamma, and the Tweedie exponential family of order 3=2.
TL;DR: In this paper, a transmuted generalized inverted exponential distribution (GIN) was proposed to generalize the two-parameter GIN using the quadratic rank transmutation map proposed by Shaw et al. (2007).
Abstract: This paper introduces a transmuted generalized inverted exponential distribution. We generalize the two parameter generalized inverted exponential distribution using the quadratic rank transmutation map proposed by Shaw et al. (2007) to develop a transmuted generalized inverted exponential distribution. The properties of the transmuted generalized inverted exponential distribution are discussed. We derive the moments and examine the order statistics. Moreover, the maximum likelihood estimators for the parameters is briefly investigated and the information matrix is derived.
TL;DR: This paper introduces a new four-parameter generalized version of the transmuted generalized linear exponential distribution, the TGLED, and provides a comprehensive account of the mathematical properties of the new distributions.
Abstract: The linear exponential distribution is a very well-known distribution for modeling lifetime data in reliability and medical studies.We introduce in this paper a new four-parameter generalized version of the transmuted generalized linear exponential distribution.We provide a comprehensive account of the mathematical properties of the new distributions. In particular, A closed-form expressions for the density, cumulative distribution ,quantile and median of the distribution is given. Also, the rth order moment and moment generating function are derived. The maximum likelihood estimation of the unknown parameters is discussed. Real data are used to determine whether the TGLED is better than other well-known distributions in modeling lifetime data or not.
Journal Article•
TL;DR: In this article, a generalized exponential power distribution (GEPD) is proposed to model the tail behavior of the distribution, which makes it more flexible and suitable for modeling than the usual normal distribution, while retaining sym- metry.
Abstract: In this paper, we propose to study a generalized form of the exponential power distribution which contains others in the literature as special cases. This unifying exponential power distribution is charac- terized by a parameter ω and a function h(ω) which regulates the tail behavior of the distribution, thus making it more flexible and suitable for modeling than the usual normal distribution, while retaining sym- metry. We derive several mathematical and statistical properties of this distribution and estimate the parameters using both the moments and maximum likelihood approach, obtaining the information matrix in the process. The multivariate extension of the distribution is also examined. Finally we fit the univariate generalized exponential power distribution as well as the normal distribution to data on eggs produced by chicken on each of two different poultry feeds (inorganic and organic copper-salt compositions) and show that the generalized exponential power distri- bution fit is considerably better. We then use the Kolmogorov-Smirnov two samples one-tailed test to show that there is an increase in egg weights and decrease in cholesterol level when the feed is organic.
TL;DR: In this article, an analytical expression for the one dimensional probability distribution of the first inter-arrival time is obtained as a solution to a system of recursive differential equations, and an extension is provided for the power series expansion of the geometric renewal function in the case of the Weibull distribution.
03 Jul 2013
TL;DR: A novel algorithm to learn mixtures of Gamma distributions with a fixed rate parameter is introduced, which converges locally and is computationally faster than an Expectation-Maximization method for Gamma mixture models.
Abstract: We introduce a novel algorithm to learn mixtures of Gamma distributions. This is an extension of the k-Maximum Likelihood Estimator algorithm for mixtures of exponential families. Although Gamma distributions are exponential families, we cannot rely directly on the exponential families tools due to the lack of closed-form formula and the cost of numerical approximation: our method uses Gamma distributions with a fixed rate parameter and a special step to choose this parameter is added in the algorithm. Since it converges locally and is computationally faster than an Expectation-Maximization method for Gamma mixture models, our method can be used beneficially as a drop-in replacement in any application using this kind of statistical models.
TL;DR: In this paper, another version of the generalized exponential geometric distribution different to that of Silva et al. (2010) is proposed, which is a three-parameter lifetime distribution with decreasing, increasing, and bathtub failure rate function.
Abstract: In this article, another version of the generalized exponential geometric distribution different to that of Silva et al. (2010) is proposed. This new three-parameter lifetime distribution with decreasing, increasing, and bathtub failure rate function is created by compounding the generalized exponential distribution of Gupta and Kundu (1999) with a geometric distribution. Some basic distributional properties, moment-generating function, rth moment, and Renyi entropy of the new distribution are studied. The model parameters are estimated by the maximum likelihood method and the asymptotic distribution of estimators is discussed. Finally, an application of the new distribution is illustrated using the two real data sets.
TL;DR: In this article, a Bayesian analysis for beta generalized distributions and related exponentiated models is presented. But this analysis is restricted to a real data set and is not applied to real-world data.
Abstract: We introduce a Bayesian analysis for beta generalized distributions and related exponentiated models. We review the exponentiated exponential, exponentiated Weibull and beta generalized exponential distributions. These distributions have been proposed as alternative extensions of the gamma and Weibull distributions in the analysis of lifetime data. Some posterior summaries of interest are obtained using Monte Carlo Markov chain (MCMC) methods. An application to a real data set is given to illustrate the potentiality of the Bayesian analysis.