Mathematica Slovaca (01399918)71(6)pp. 1581-1598
The main aim of this paper is to introduce a new class of continuous generalized exponential distributions, both for the univariate and bivariate cases. This new class of distributions contains some newly developed distributions as special cases, such as the univariate and also bivariate geometric generalized exponential distribution and the exponential-discrete generalized exponential distribution. Several properties of the proposed univariate and bivariate distributions, and their physical interpretations, are investigated. The univariate distribution has four parameters, whereas the bivariate distribution has five parameters. We propose to use an EM algorithm to estimate the unknown parameters. According to extensive simulation studies, we see that the effectiveness of the proposed algorithm, and the performance is quite satisfactory. A bivariate data set is analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice. © 2021 Mathematical Institute Slovak Academy of Sciences.
Sequential Analysis (07474946)38(3)pp. 279-300
In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated. © 2019, © 2019 Taylor & Francis Group, LLC.
Communications in Statistics - Theory and Methods (1532415X)48(14)pp. 3464-3481
Recently, Lee and Cha proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall–Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper. © 2018, © 2018 Taylor & Francis Group, LLC.
Journal of Statistical Theory and Practice (15598616)12(3)pp. 595-614
In 1997, Marshall and Olkin introduced a very powerful method to introduce an additional parameter to a class of continuous distribution functions that brings more flexibility to the model. They demonstrated their method for the exponential and Weibull classes. In the same paper they briefly indicated its bivariate extension. The main aim of this article is to introduce the same method, for the first time, to the class of discrete generalized exponential distributions both for the univariate and bivariate cases. We investigate several properties of the proposed univariate and bivariate classes. The univariate class has three parameters, whereas the bivariate class has five parameters. It is observed that depending on the parameter values, the univariate class can be zero inflated as well as heavy tailed. We propose to use an expectation–maximization (EM) algorithm to estimate the unknown parameters. Small simulation experiments have been performed to see the effectiveness of the proposed EM algorithm, and a bivariate data set has been analyzed; it is observed that the proposed models and the EM algorithm work quite well in practice. © 2018, © 2018 Grace Scientific Publishing, LLC.
Communications in Statistics - Theory and Methods (1532415X)46(9)pp. 4296-4310
In this paper, the researchers attempt to introduce a new generalization of the Weibull-geometric distribution. The failure rate function of the new model is found to be increasing, decreasing, upside-down bathtub, and bathtub-shaped. The researchers obtained the new model by compounding Weibull distribution and discrete generalized exponential distribution of a second type, which is a generalization of the geometric distribution. The new introduced model contains some previously known lifetime distributions as well as a new one. Some basic distributional properties and moments of the new model are discussed. Estimation of the parameters is illustrated and the model with two known real data sets is examined. © 2017 Taylor & Francis Group, LLC.
Statistics (02331888)51(5)pp. 1143-1158
In this paper, we develop a bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized exponential distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate distribution is a very flexible distribution and the bivariate geometric distribution can be obtained as a special case of this distribution. The proposed distribution can be seen as a natural discrete analogue of the bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally, we propose some open problems and conclude the paper. © 2017 Informa UK Limited, trading as Taylor & Francis Group.
Communications in Statistics - Theory and Methods (1532415X)45(5)pp. 1575-1575
Hacettepe Journal of Mathematics and Statistics (2651477X)45(6)pp. 1767-1779
In this paper, a new three-parameter extension of the generalized geo- metric distribution of [6] is introduced. The new discrete distribution belongs to the resilience parameter family and handles a decreasing, in- creasing, upside-down and bathtub-shaped hazard rate function. The new distributions can also be considered as discrete analogs of some recent continuous distributions belonging to the known Marshall-Olkin family. Here, some basic statistical and mathematical properties of the new distribution are studied. In addition, estimation of the unknown parameters, a simulated example and an application of the new model are illustrated. © 2016, Hacettepe University. All rights reserved.
Journal of Applied Statistics (02664763)42(12)pp. 2654-2670
In this paper, a discrete counterpart of the general class of continuous beta-G distributions is introduced. A discrete analog of the beta generalized exponential distribution of Barreto-Souza et al. [2], as an important special case of the proposed class, is studied. This new distribution contains some previously known discrete distributions as well as two new models. The hazard rate function of the new model can be increasing, decreasing, bathtub-shaped and upside-down bathtub. Some distributional and moment properties of the new distribution as well as its order statistics are discussed. Estimation of the parameters is illustrated using the maximum likelihood method and, finally, the model with a real data set is examined. © 2015 Taylor & Francis.
Communications in Statistics - Theory and Methods (1532415X)44(10)pp. 2079-2091
There are not many known distributions for modeling discrete data. In this paper, we shall introduce a discrete analogue of the beta-exponential distribution of Nadarajah and Kotz (2006), which is more plausible in modeling discrete data and exhibits both increasing and decreasing hazard rates. The discrete beta-exponential distribution can be viewed as a generalization of the discrete generalized exponential distribution introduced by Nekoukhou et al. (2012) and, thus, as an another generalization of the geometric distribution. We shall first study some basic distributional and moment properties of the new distribution. Then, certain structural properties of the distribution such as its unimodality, hazard rate behavior, and Rényi entropy are discussed. Using the maximum likelihood method, estimation of the model parameters is also investigated. Finally, the model is examined with a real data set and compared with its rival model, that is, the discrete generalized exponential distribution. 2015 Copyright © Taylor & Francis Group, LLC.
Communications in Statistics Part B: Simulation and Computation (15324141)44(6)pp. 1389-1404
In this article, the exponentiated Weibull distribution is extended by the Marshall-Olkin family. Our new four-parameter family has a hazard rate function with various desired shapes depending on the choice of its parameters and, thus, it is very flexible in data modeling. It also contains two mixed distributions with applications to series and parallel systems in reliability and also contains several previously known lifetime distributions. We shall study some basic distributional properties of the new distribution. Some closed forms are derived for its moment generating function and moments as well as moments of its order statistics. The model parameters are estimated by the maximum likelihood method. The stress-strength parameter and its estimation are also investigated. Finally, an application of the new model is illustrated using two real datasets. © 2015 Taylor & Francis Group, LLC.
SORT (16962281)39(1)pp. 127-146
In this paper, the exponentiated discrete Weibull distribution is introduced. This new generalization of the discrete Weibull distribution can also be considered as a discrete analog of the exponentiated Weibull distribution. A special case of this exponentiated discrete Weibull distribution defines a new generalization of the discrete Rayleigh distribution for the first time in the literature. In addition, discrete generalized exponential and geometric distributions are some special sub-models of the new distribution. Here, some basic distributional properties, moments, and order statistics of this new discrete distribution are studied. We will see that the hazard rate function can be increasing, decreasing, bathtub, and upside-down bathtub shaped. Estimation of the parameters is illustrated using the maximum likelihood method. The model with a real data set is also examined.
Communications in Statistics - Theory and Methods (1532415X)42(13)pp. 2324-2334
Skew-symmetric distributions of various types have been the center of attraction by many researchers in the literature. In this article, we will introduce a uni/bimodal generalization of the Azzalini's skew-normal distribution which is indeed an extension of the skew-generalized normal distribution obtained by Arellano-Valle et al. (2004). Our new distribution contains more parameters and thus it is more flexible in data modeling. Indeed, certain univariate case of the so called flexible skew-symmetric distribution of Ma and Genton (2004) is also a particular case of our proposed model. We will first study some basic distributional properties of the new extension, such as its distribution function, limiting behavior and moments. Then, we will investigate some useful results regarding its relation with other known distributions, such as student's t and skew-Cauchy distributions. In addition, we will present certain methods to generate the new distribution and, finally, we shall apply the model to a real data set to illustrate its behavior comparing to some rival models. © Taylor and Francis Group, LLC.
Statistics (02331888)47(4)pp. 876-887
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 will investigate estimation of the parameters involved. Finally, we will examine the model with a real data set. © 2013 Copyright Taylor and Francis Group, LLC.
SORT (16962281)37(2)pp. 211-230
In this paper, we will introduce the new Kumaraswamy-power series class of distributions. This new class is obtained by compounding the Kumaraswamy distribution of Kumaraswamy (1980) and the family of power series distributions. The new class contains some new double bounded distributions such as the Kumaraswamy-geometric,-Poisson,-logarithmic and-binomial, which are used widely in hydrology and related areas. In addition, the corresponding hazard rate function of the new class can be increasing, decreasing, bathtub and upside-down bathtub. Some basic properties of this class of distributions such as the moment generating function, moments and order statistics are studied. Some special members of the class are also investigated in detail. The maximum likelihood method is used for estimating the unknown parameters of the members of the new class. Finally, an application of the proposed class is illustrated using a real data set.
Communications in Statistics - Theory and Methods (1532415X)41(11)pp. 2000-2013
In this article, we attempt to introduce a discrete analog of the generalized exponential distribution of Gupta and Kundu (1999). This new discrete generalized exponential (DGE(, p)) distribution can be viewed as another generalization of the geometric distribution and it is more flexible in data modeling. We shall first study some basic distributional and moment properties of this family of new distributions. Then, we will reveal their structural properties and applications and also investigate estimation of their parameters. Finally, we shall discuss their convolution properties and arrive at some characterizations in the special cases DGE(2, p) and DGE(3, p). © 2012 Taylor and Francis Group, LLC.
Statistical Papers (09325026)53(3)pp. 685-696
Skew-symmetric distributions of various types have been the center of attraction by many researchers in the literature. In this article, we shall introduce another more general class of skew distributions, specially related to the Laplace distribution. This new class contains some previously known skew distributions. We shall investigate different characteristics of members of this class such as its moments, thus generalizing a result of Umbach (Stat Probab Lett 76:507-512, 2006), limiting behavior, moment generating function, unimodality and reveal its natural occurrence as the distribution of some order statistics. In addition, we will generalize a result of Aryal and Rao (Nonlinear Anal 63:639-646, 2005) in connection with truncated skew-Laplace distribution and study its certain stochastic orderings. Some illustrative examples are also provided. © 2011 Springer-Verlag.
