Showing papers on "Natural exponential family published in 2014"
Book•
05 May 2014TL;DR: In this article, the information and exponential families in statistical theory were studied. But they did not consider the exponential family in the context of exponential families. And they were not considered in this paper.
Abstract: (1980). Information and Exponential Families in Statistical Theory. Technometrics: Vol. 22, No. 2, pp. 280-280.
717 citations
TL;DR: In this paper, the authors proposed the use of quantile functions to define the W function of the T-X family of distributions, which provides a new method of generating univariate distributions.
Abstract: The cumulative distribution function (CDF) of the T-X family is given by R{W(F(x))}, where R is the CDF of a random variable T, F is the CDF of X and W is an increasing function defined on [0, 1] having the support of T as its range. This family provides a new method of generating univariate distributions. Different choices of the R, F and W functions naturally lead to different families of distributions. This paper proposes the use of quantile functions to define the W function. Some general properties of this T-X system of distributions are studied. It is shown that several existing methods of generating univariate continuous distributions can be derived using this T-X system. Three new distributions of the T-X family are derived, namely, the normal-Weibull based on the quantile of Cauchy distribution, normal-Weibull based on the quantile of logistic distribution, and Weibull-uniform based on the quantile of log-logistic distribution. Two real data sets are applied to illustrate the flexibility of the distributions.
113 citations
TL;DR: In this article, the authors proposed four families of generalized normal distributions arising from the quantile functions of (i) standard exponential, (ii) standard log-logistic, (iii) normal logistic and (iv) standard extreme value distributions.
Abstract: The idea of generating skewed distributions from normal has been of great interest among researchers for decades. This paper proposes four families of generalized normal distributions using the T-X framework. These four families of distributions are named as T-normal families arising from the quantile functions of (i) standard exponential, (ii) standard log-logistic, (iii) standard logistic and (iv) standard extreme value distributions. Some general properties including moments, mean deviations and Shannon entropy of the T-normal family are studied. Four new generalized normal distributions are developed using the T-normal method. Some properties of these four generalized normal distributions are studied in detail. The shapes of the proposed T-normal distributions can be symmetric, skewed to the right, skewed to the left, or bimodal. Two data sets, one skewed unimodal and the other bimodal, are fitted by using the generalized T-normal distributions. 60E05; 62E15; 62P10
85 citations
TL;DR: In this paper, a new generalization of the complementary Weibull geometric distribution (CWGD) was proposed, referred to as the transmuted complementary weibull geodesic distribution (TCWGD), which includes as special cases the complementary CWGD, complementary exponential geometric distribution(CEGD), Weibell distribution(WD), and exponential distribution (ED).
Abstract: This paper provides a new generalization of the complementary Weibull geometric distribution that introduced by Tojeiro et al. (2014), using the quadratic rank transmutation map studied by Shaw and Buckley (2007). The new distribution is referred to as transmuted complementary Weibull geometric distribution (TCWGD). The TCWG distribution includes as special cases the complementary Weibull geometric distribution (CWGD), complementary exponential geometric distribution(CEGD),Weibull distribution (WD) and exponential distribution (ED). Various structural properties of the new distribution including moments, quantiles, moment generating function and ROnyi entropy of the subject distribution are derived. We proposed the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. A real data set are used to compare the exibility of the transmuted version versus the complementary Weibull geometric distribution.
56 citations
TL;DR: A transmuted linear exponential distribution is developed that generalizes thelinear exponential distribution with an additional parameter using the quadratic rank transmutation map which was studied by Shaw et al.
Abstract: In this article, a transmuted linear exponential distribution is developed that generalizes the linear exponential distribution with an additional parameter using the quadratic rank transmutation map which was studied by Shaw et al. Some statistical properties of the proposed distribution such as moments, quantiles, and the failure rate function are investigated. The maximum likelihood estimators of unknown parameters are also discussed and a real data analysis is carried out to illustrate the superiority of the proposed distribution.
55 citations
29 Sep 2014
TL;DR: In this article, the authors discuss the properties of mixed exponential distributions and consider their examples and variations appearing often in insurance and discuss their properties and their variations appearing in insurance policies, including the Laplace transform and the Pareto distribution.
Abstract: This articles discusses the properties mixed exponential distributions and consider their examples and variations appearing often in insurance.
Keywords:
exponential distribution;
mixture of exponential distributions;
Laplace transform;
hyperexponential distribution;
Pareto distribution;
log-normal distribution;
ruin probability;
decreasing failure rate;
infinitely divisible;
Tauberian theorem;
frailty model
40 citations
29 Sep 2014
TL;DR: In this article, the skew-normal family of distributions is introduced, both in the univariate and the multivariate setting, and its main properties are summarized, and a number of extensions are also presented, from the simplest forms denoted as "extended skew normal" to more complex ones, such as skew-elliptical and skew-symmetric distributions.
Abstract: The skew-normal family of distributions is introduced, both in the univariate and the multivariate setting, and its main properties are summarized. A number of extensions are also presented, from the simplest forms denoted “extended skew-normal” to more complex ones, such as the skew-elliptical and skew-symmetric distributions.
Keywords:
asymmetric distributions;
probability distributions;
multivariate statistics;
elliptical distributions
39 citations
TL;DR: In this paper, the authors introduced a new class of models called the Marshall-Olkin extended Weibull family of distributions based on the work by Marshall and Olkin (Biometrika 84:641-652, 1997).
Abstract: We introduce a new class of models called the Marshall-Olkin extended Weibull family of distributions based on the work by Marshall and Olkin (Biometrika 84:641–652, 1997). The proposed family includes as special cases several models studied in the literature such as the Marshall-Olkin Weibull, Marshall-Olkin Lomax, Marshal-Olkin Frechet and Marshall-Olkin Burr XII distributions, among others. It defines at least twenty-one special models and thirteen of them are new ones. We study some of its structural properties including moments, generating function, mean deviations and entropy. We obtain the density function of the order statistics and their moments. Special distributions are investigated in some details. We derive two classes of entropy and one class of divergence measures which can be interpreted as new goodness-of-fit quantities. The method of maximum likelihood for estimating the model parameters is discussed for uncensored and multi-censored data. We perform a simulation study using Markov Chain Monte Carlo method in order to establish the accuracy of these estimators. The usefulness of the new family is illustrated by means of two real data sets. 60E05; 62F03; 62F10; 62P10
30 citations
TL;DR: This paper uses the GIG distribution in the context of the scale-mixture of skew-normal distributions, deriving a new family of distributions called Skew-Normal Generalized Hyperbolic distributions, which possesses skewness with heavy-tails, and generalizes the symmetric normal inverse Gaussian and symmetric generalized hyperbolic distribution.
28 citations
TL;DR: This paper investigates a problem of exponential state estimation for Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays with delay-distribution-dependent exponential stability criteria obtained in terms of linear matrix inequalities.
Abstract: In this paper, we investigate a problem of exponential state estimation for Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays. A new type of mode-dependent probabilistic leakage time-varying delay is considered. Given the probability distribution of the time-delays, stochastic variables that satisfying Bernoulli random binary distribution are formulated to produce a new system which includes the information of the probability distribution. Under these circumstances, the state estimator is designed to estimate the true concentration of the mRNA and the protein of the GRNs. Based on Lyapunov–Krasovskii functional that includes new triple integral terms and decomposed integral intervals, delay-distribution-dependent exponential stability criteria are obtained in terms of linear matrix inequalities. Finally, a numerical example is provided to show the usefulness and effectiveness of the obtained results.
TL;DR: This article studies the optimal constant-stress accelerated life tests with complete sample for the generalized exponential distribution using V-optimality as well as D- Optimality criteria to reach interesting conclusions.
Abstract: Recently generalized exponential distribution has been discussed by many authors. In this article, we study the optimal constant-stress accelerated life tests with complete sample for the generalized exponential distribution. The problem of choosing the optimal proportions of test units allocated to each stress level is addressed by using V-optimality as well as D-optimality criteria. Some interesting conclusions are obtained. Finally, real data example and numerical examples have been analyzed to illustrate the proposed procedures.
TL;DR: In this paper, a new two-parameter lifetime distribution with increasing failure rate was proposed, which is constructed as a distribution of a random sum of independent exponential random variables when the sample size has a zero truncated binomial distribution.
01 Jan 2014
TL;DR: The gamma-inverse Weibull (GIW) distribution as mentioned in this paper is a special case of the standard W-distribution and is useful for failure time data analysis, however, it is not suitable for real data sets.
Abstract: The gamma-inverse Weibull (GIW) distribution which includes in- verse Weibull, inverse exponential, gamma-inverse exponential, gamma- inverse Rayleigh, inverse Rayleigh, gamma-Frechet and Frechet distri- butions as special cases is proposed and studied. This new distribu- tion might be useful for failure time data analysis. Some mathematical properties of the new distribution including moments, mean deviations, Bonferroni and Lorenz curves, Shannon and Renyi entropies are pre- sented. Maximum likelihood estimation technique is used to estimate the parameters and applications to real data sets are given to illustrate the usefulness of this new class of distributions.
Journal Article•
TL;DR: In this article, a three-parameter Kumaraswamy-Inverse Exponential distribution is proposed and some of its statistical properties are identified, which is a viable alternative to the beta distribution.
Abstract: The Kumaraswamy distribution being a viable alternative to the beta distribution is being used to propose a three-parameter Kumaraswamy-Inverse Exponential distribution and some of its statistical properties are identified.
30 Sep 2014
TL;DR: In this article, the adaptive type-II progressive hybrid censoring scheme (AT-II PHCS) in the presence of the competing risks model is considered and the maximum likelihood method is used to derive point and asymptotic confidence intervals for the unknown parameters.
Abstract: This paper presents estimates of the parameters based on adaptive type-II progressive hybrid censoring scheme (AT-II PHCS) in the presence of the competing risks model. We consider the competing risks have generalized exponential distributions (GED). The maximum likelihood method is used to derive point and asymptotic confidence intervals for the unknown parameters. The relative risks due to each cause of failure are investigated. A real data set is used to illustrate the theoretical results and to test the hypothesis that the causes of failure follow the generalized exponential distributions against the exponential distribution (ED). Keywords : Competing Risks; Adaptive Type-II Progressive Hybrid Censoring; Generalized Exponential Distribution; Maximum Likelihood Estimation.
TL;DR: In this paper, the problem of predicting the future sequential order statistics based on observed multiply Type-II censored samples from one-and two-parameter exponential distributions is addressed using the Bayesian approach, the predictive and survival functions are derived and then the point and interval predictions are obtained.
Abstract: In this paper, the problem of predicting the future sequential order statistics based on observed multiply Type-II censored samples of sequential order statistics from one- and two-parameter exponential distributions is addressed. Using the Bayesian approach, the predictive and survival functions are derived and then the point and interval predictions are obtained. Finally, two numerical examples are presented for illustration.
TL;DR: A new family of Marshall–Olkin extended distributions is introduced, discussed and analyzed and it is shown that the new family provides a better fit than some other known distributions.
Book•
15 Jan 2014
TL;DR: In this paper, the authors describe the fit of a BHS distribution to a NEF-GHS or Meixner distribution family and the BHS Distribution Family and the SHS and SASHS distribution families.
Abstract: Preface.- Hyperbolic Secant Distributions.- The GSH Distribution Family and Skew Versions.- The NEF-GHS or Meixner Distribution Family.- The BHS Distribution Family.- The SHS and SASHS Distribution Family.- Application to Finance.- R-Code: Fitting a BHS Distribution.
Journal Article•
TL;DR: In this paper, a generalized EME (GEME) distribution was proposed and various properties of the distribution were developed. But, the distribution is not suitable for the analysis of the hazard function of the EME random variable.
Abstract: Moment distributions have a vital role in mathematics and statistics, in particular in probability theory, in the perspective research related to ecology, reliability, biomedical field, econometrics, survey sampling and in life-testing. Hasnain (2013) developed an exponentiated moment exponential (EME) distribution and discussed some of its important properties. In the present work, we propose a generalization of EME distribution which we call it generalized EME (GEME) distribution and develop various properties of the distribution. We also present characterizations of the distribution in terms of conditional expectation as well as based on hazard function of the GEME random variable.
Posted Content•
TL;DR: In this paper, a numerical method to approximate ruin probabilities is proposed within the frame of a compound Poisson ruin model, where the defective density function associated to the ruin probability is projected in an orthogonal polynomial system.
Abstract: A numerical method to approximate ruin probabilities is proposed within the frame of a compound Poisson ruin model. The defective density function associated to the ruin probability is projected in an orthogonal polynomial system. These polynomials are orthogonal with respect to a probability measure that belongs to Natural Exponential Family with Quadratic Variance Function (NEF-QVF). The method is convenient in at least four ways. Firstly, it leads to a simple analytical expression of the ultimate ruin probability. Secondly, the implementation does not require strong computer skills. Thirdly, our approximation method does not necessitate any preliminary discretisation step of the claim sizes distribution. Finally, the coefficients of our formula do not depend on initial reserves.
TL;DR: This paper derives expressions for the probability mass functions of geometric and binomial distributions of order k using Bernoulli trials with a geometrically varying success probability.
Journal Article•
TL;DR: In this article, a generalization of the inverted exponential distribution is proposed, which serves as a competitive model and an alternative to both the generalized inverse exponential distribution and the inverse exponential distributions.
Abstract: We provide another generalization of the inverted exponential distribution which serves as a competitive model and an alternative to both the generalized inverse exponential distribution and the inverse exponential distribution. The model is positively skewed and its shape could be decreasing or unimodal (depending on its parameter values). The statistical properties of the proposed model are provided and the method of Maximum Likelihood Estimation (MLE) was proposed in estimating its parameters.
TL;DR: In this article, a class of absolutely continuous bivariate exponential distributions is constructed using the product form of a first order autoregressive model, and inference methods are proposed for parameter estimation and diagnosis.
TL;DR: A new family of distributions called the gamma extended Weibull family is introduced, which includes several well-known models as special cases and defines at least seventeen new special models.
Abstract: We introduce a new family of distributions called the gamma extended Weibull family. The proposed family includes several well-known models as special cases and defines at least seventeen new special models. Structural properties of this family are studied. Additionally, the maximum likelihood method for estimating the model parameters is discussed. An application to real data illustrates the usefulness of the new family. The results provide evidence that the proposed family outperforms other classes of lifetime models.
TL;DR: The Libby-Novick beta family as mentioned in this paper is a family of distributions, which includes the classical beta generalized and exponentiated generators, which can control skewness and kurtosis simultaneously, vary tail weights and provide more entropy for generated distributions.
Abstract: We define a family of distributions, named the Libby-Novick beta family of distributions, which includes the classical beta generalized and exponentiated generators. The new family offers much more flexibility for modeling real data than these two generators and the Kumaraswamy family of distributions (Cordeiro \& de Castro, 2011). The extended fa\-mily provides reasonable parametric fits to real data in several areas because the additional shape parameters can control the skewness and kurtosis simultaneously, vary tail weights and provide more entropy for the generated distribution. For any given distribution, we can construct a wider distribution with three additional shape parameters which has much more flexibility than the original one. The family density function is a linear combination of exponentiated densities defined from the same baseline distribution. The proposed family also has tractable mathematical properties including moments, generating function, mean deviations and order statistics. The parameters are estimated by maximum likelihood and the observed information matrix is determined. The importance of the family is very well illustrated. By means of two real data sets we demonstrate that this family can give better fits than those ones using the McDonald, beta generalized and Kumaraswamy generalized classes of distributions.
TL;DR: In this article, a novel approach to data analysis using fractional order calculus is presented, which can be applied to any distribution and shows remarkable improvement even if the parameters of a particular distribution have been optimised to achieve the best fit to data.
Journal Article•
TL;DR: In this article, a new class of length-biased of weighted exponential and Rayleigh distributions (LBW 1 E 1 D), (LBWRD) is introduced. But, it is not suitable for the use of lifetime data.
Abstract: The concept of length-biased distribution can be employed in development of proper models for lifetime data. Length-biased distribution is a special case of the more general form known as weighted distribution. In this paper we introduce a new class of length-biased of weighted exponential and Rayleigh distributions(LBW 1 E 1 D), (LBWRD).This paper surveys some of the possible uses of Length - biased distribution We study the some of its statistical properties with application of these new distribution . Keywords : length- biased weighted Rayleigh distribution, length- biased weighted exponential distribution, maximum likelihood estimation.
TL;DR: The Hyperbolic-Secant (HS) distribution as mentioned in this paper is the generator distribution of the sixth natural exponential family with quadratic variance function, but it is much less known than other distributions in the exponential families.
Abstract: Although it is the generator distribution of the sixth natural exponential family with quadratic variance function, the Hyperbolic-Secant (HS) distribution is much less known than other distributions in the exponential families. Its lack of familiarity is due to its isolation from many widely used statistical models. We fill in the gap by showing three examples naturally generating the HS distribution, including Fisher’s analysis of similarity between twins, the Jeffreys’ prior for contingency tables, and invalid instrumental variables.
TL;DR: In this article, a generalized linear exponential distribution (NGLED) is considered, which can be considered as a new and more flexible extension of linear exponential distributions, and some statistical properties for the NGLED such as the hazard rate function, moments, quantiles are given.
Abstract: A new generalized linear exponential distribution (NGLED) is considered in this paper which can be deemed as a new and more flexible extension of linear exponential distribution. Some statistical properties for the NGLED such as the hazard rate function, moments, quantiles are given. The maximum likelihood estimations (MLE) of unknown parameters are also discussed. A simulation study and two real data analyzes are carried out to illustrate that the new distribution is more flexible and effective than other popular distributions in modeling lifetime data.