An extension of the generalized linear failure rate distribution

In this article, we introduce a new extension of the generalized linear failure rate (GLFR) distributions. It includes some well-known lifetime distributions such as extension of generalized exponential and GLFR distributions as special sub-models. In addition, it can have a constant, decreasing, increasing, upside-down bathtub (unimodal), and bathtub-shaped hazard rate function (hrf) depending on its parameters. We provide some of its statistical properties such as moments, quantiles, skewness, kurtosis, hrf, and reversible hrf. The maximum likelihood estimation of the parameters is also discussed. At the end, a real dataset is given to illustrate the usefulness of this new distribution in analyzing lifetime data.     

A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.     

Exponentiated power Lindley power series class of distributions: Theory and applications

ABSTRACT

We introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley power series (EPLPS) distribution. The new distribution arises on a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the minimum lifetime value among all risks. The distribution exhibits a variety of bathtub-shaped hazard rate functions. It contains as particular cases several lifetime distributions. Various properties of the distribution are investigated including closed-form expressions for the density function, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Expressions for the Rényi and Shannon entropies are also given. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Finally, two data applications are given showing flexibility and potentiality of the EPLPS distribution.     

On the failure rate estimation of the inverse gaussian distribution

New estimators of the inverse Gaussian failure rate are proposed based on the maximum likelihood predictive densities derived by Yang (1999). These estimators are compared, via Monte Carlo simulation, with the usual maximum likelihood estimators of the failure rate and found to be superior in terms of bias and mean squared error. Sensitivity of the estimators against the departure from the inverse Gaussian distribution is studied.     

A new two-parameter lifetime distribution with decreasing,increasing or upside-down bathtub-shaped failure rate

In this paper, we introduce a generalization of the Bilal distribution, where a new two-parameter distribution is presented. We show that its failure rate function can be upside-down bathtub shaped. The failure rate can also be decreasing or increasing. A comprehensive mathematical treatment of the new distribution is provided. The estimation by maximum likelihood is discussed, and a closed-form expression for Fisher’s information matrix is obtained. Asymptotic interval estimators for both of the two unknown parameters are also given. A simulation study is conducted and applications to real data sets are presented.     

The power series skew normal class of distributions

A class of power series skew normal distributions is introduced by generalizing the geometric skew normal distribution of Kundu. Various mathematical properties are derived and estimation addressed by the method of maximum likelihood. The data application of Kundu [Sankhyā B, 76, 2014, 167–189] is revisited and the proposed class is shown to provide a better fit.     

On the mixtures of Weibull and Pareto (IV) distribution: An alternative to Pareto distribution

Finite mixture models have provided a reasonable tool to model various types of observed phenomena, specially those which are random in nature. In this article, a finite mixture of Weibull and Pareto (IV) distribution is considered and studied. Some structural properties of the resulting model are discussed including estimation of the model parameters via expectation maximization (EM) algorithm. A real-life data application exhibits the fact that in certain situations, this mixture model might be a better alternative than the rival popular models.     

A new lifetime model with different types of failure rate

A new class of lifetime distributions, which can exhibit with upside-down bathtub-shaped, bathtub-shaped, decreasing, and increasing failure rates, is introduced. The new distribution is constructed by compounding generalized Weibull and logarithmic distributions, leading to improvement on the lifetime distribution considered in Dimitrakopoulou et al. (2007 Dimitrakopoulou, T., K. Adamidis, and S. Loukas. 2007. A lifetime distribution with an upside-down bathtub-shaped hazard function. IEEE Transactions on Reliability 56:308–11.[Crossref], [Web of Science ®] , [Google Scholar]) by having no restriction on the shape parameter and extending the result studied by Tahmasbi and Rezaei (2008 Tahmasbi, R., and S. Rezaei. 2008. A two-parameter lifetime distribution with decreasing failure rate. Computational Statistics and Data Analysis 52:3889–901.[Crossref], [Web of Science ®] , [Google Scholar]) in the general form. The proposed model includes the exponential–logarithmic and Weibull–logarithmic distributions as special cases. Various statistical properties of the proposed class are discussed. Furthermore, estimation via the maximum likelihood method and the Fisher information matrix are discussed. Applications to real data demonstrate that the new class of distributions is more flexible than other recently proposed classes.     

Semiparametric estimation of the cumulative incidence function under competing risks and left-truncated sampling

In this article, we propose semiparametric methods to estimate the cumulative incidence function of two dependent competing risks for left-truncated and right-censored data. The proposed method is based on work by Huang and Wang (1995). We extend previous model by allowing for a general parametric truncation distribution and a third competing risk before recruitment. Based on work by Vardi (1989), several iterative algorithms are proposed to obtain the semiparametric estimates of cumulative incidence functions. The asymptotic properties of the semiparametric estimators are derived. Simulation results show that a semiparametric approach assuming the parametric truncation distribution is correctly specified produces estimates with smaller mean squared error than those obtained in a fully nonparametric model.     

Generalized inverse Lindley distribution with application to Danish fire insurance data

The Danish fire insurance data have recently been modeled by composite distributions, i.e., distributions made up by piecing together two or more distributions. Here, we introduce a new non composite distribution that performs well with respect to the Danish fire insurance data. It fits better than almost all of the commonly known heavy-tailed distributions and some of the composite distributions.     

A new flexible hazard rate distribution: Application and estimation of its parameters

We introduce a new class of flexible hazard rate distributions which have constant, increasing, decreasing, and bathtub-shaped hazard function. This class of distributions obtained by compounding the power and exponential hazard rate functions, which is called the power-exponential hazard rate distribution and contains several important lifetime distributions. We obtain some distributional properties of the new family of distributions. The estimation of parameters is obtained by using the maximum likelihood and the Bayesian methods under squared error, linear-exponential, and Stein’s loss functions. Also, approximate confidence intervals and HPD credible intervals of parameters are presented. An application to real dataset is provided to show that the new hazard rate distribution has a better fit than the other existing hazard rate distributions and some four-parameter distributions. Finally , to compare the performance of proposed estimators and confidence intervals, an extensive Monte Carlo simulation study is conducted.     

Missing covariates in generalized linear models when the missing data mechanism is non-ignorable

We propose a method for estimating parameters in generalized linear models with missing covariates and a non-ignorable missing data mechanism. We use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one-dimensional conditional distributions, with each one-dimensional conditional distribution consisting of a logistic regression. We allow the covariates to be either categorical or continuous. The joint covariate distribution is also modelled via a sequence of one-dimensional conditional distributions, and the response variable is assumed to be completely observed. We derive the E- and M-steps of the EM algorithm with non-ignorable missing covariate data. For categorical covariates, we derive a closed form expression for the E- and M-steps of the EM algorithm for obtaining the maximum likelihood estimates (MLEs). For continuous covariates, we use a Monte Carlo version of the EM algorithm to obtain the MLEs via the Gibbs sampler. Computational techniques for Gibbs sampling are proposed and implemented. The parametric form of the assumed missing data mechanism itself is not `testable' from the data, and thus the non-ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model. Therefore, although a model may have `passed' the tests for a certain missing data mechanism, this does not mean that we have captured, even approximately, the correct missing data mechanism. Hence, model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail. Several simulations are given to demonstrate the methodology. In addition, a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed.     

An EM and a Stochastic Version of the EM Algorithm for Nonparametric Hidden Semi-Markov Models

The Hidden semi-Markov models (HSMMs) were introduced to overcome the constraint of a geometric sojourn time distribution for the different hidden states in the classical hidden Markov models. Several variations of HSMMs were proposed that model the sojourn times by a parametric or a nonparametric family of distributions. In this article, we concentrate our interest on the nonparametric case where the duration distributions are attached to transitions and not to states as in most of the published papers in HSMMs. Therefore, it is worth noticing that here we treat the underlying hidden semi-Markov chain in its general probabilistic structure. In that case, Barbu and Limnios (2008 Barbu , V. , Limnios , N. ( 2008 ). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications: Their Use in Reliability and DNA Analysis . New York : Springer . [Google Scholar]) proposed an Expectation–Maximization (EM) algorithm in order to estimate the semi-Markov kernel and the emission probabilities that characterize the dynamics of the model. In this article, we consider an improved version of Barbu and Limnios' EM algorithm which is faster than the original one. Moreover, we propose a stochastic version of the EM algorithm that achieves comparable estimates with the EM algorithm in less execution time. Some numerical examples are provided which illustrate the efficient performance of the proposed algorithms.     

The compound class of linear failure rate-power series distributions: Model,properties, and applications

In this article, a new class of distributions is introduced, which generalizes the linear failure rate distribution and is obtained by compounding this distribution and power series class of distributions. This new class of distributions is called the linear failure rate-power series distributions and contains some new distributions such as linear failure rate-geometric, linear failure rate-Poisson, linear failure rate-logarithmic, linear failure rate-binomial distributions, and Rayleigh-power series class of distributions. Some former works such as exponential-power series class of distributions, exponential-geometric, exponential-Poisson, and exponential-logarithmic distributions are special cases of the new proposed model. The ability of the linear failure rate-power series class of distributions is in covering five possible hazard rate function, that is, increasing, decreasing, upside-down bathtub (unimodal), bathtub and increasing-decreasing-increasing shaped. Several properties of this class of distributions such as moments, maximum likelihood estimation procedure via an EM-algorithm and inference for a large sample, are discussed in this article. In order to show the flexibility and potentiality, the fitted results of the new class of distributions and some of its submodels are compared using two real datasets.     

A new three-parameter ageing distribution

We propose a new three-parameter ageing distribution called the Weibull-Poisson (WP) distribution, which generalizes the exponential-Poisson (EP) distribution introduced by Kus (2007). This new distribution has a more general form of failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model three ageing classes of life distributions including decreasing failure rate (DFR), increasing failure rate (IFR), and modified upside-down-bathtub (MUBT)-shaped failure rate. It thus provides an alternative to many existing life distributions. Various properties of this distribution are discussed and the estimation of the parameters is considered by the expectation maximization (EM) algorithm. Also, the asymptotic variance-covariance matrices of these estimates are obtained. Furthermore, some expressions for the Rènyi and Shannon entropies are given. Simulation studies are performed and experimental results are illustrated based on a real data set.     

Fitting mixtures of linear regressions

In most applications, the parameters of a mixture of linear regression models are estimated by maximum likelihood using the expectation maximization (EM) algorithm. In this article, we propose the comparison of three algorithms to compute maximum likelihood estimates of the parameters of these models: the EM algorithm, the classification EM algorithm and the stochastic EM algorithm. The comparison of the three procedures was done through a simulation study of the performance (computational effort, statistical properties of estimators and goodness of fit) of these approaches on simulated data sets.

Simulation results show that the choice of the approach depends essentially on the configuration of the true regression lines and the initialization of the algorithms.     

Gamma lifetimes and associated inference for interval-censored cure rate model with COM–Poisson competing cause

In this article, we consider a competing cause scenario and assume the wider family of Conway–Maxwell–Poisson (COM–Poisson) distribution to model the number of competing causes. Assuming the type of the data to be interval censored, the main contribution is in developing the steps of the expectation maximization (EM) algorithm to determine the maximum likelihood estimates (MLEs) of the model parameters. A profile likelihood approach within the EM framework is proposed to estimate the COM–Poisson shape parameter. An extensive simulation study is conducted to evaluate the performance of the proposed EM algorithm. Model selection within the wider class of COM–Poisson distribution is carried out using likelihood ratio test and information-based criteria. A study to demonstrate the effect of model mis-specification is also carried out. Finally, the proposed estimation method is applied to a data on smoking cessation and a detailed analysis of the obtained results is presented.     

Gompertz-power series distributions

ABSTRACT

In this article, we introduce the Gompertz power series (GPS) class of distributions which is obtained by compounding Gompertz and power series distributions. This distribution contains several lifetime models such as Gompertz-geometric (GG), Gompertz-Poisson (GP), Gompertz-binomial (GB), and Gompertz-logarithmic (GL) distributions as special cases. Sub-models of the GPS distribution are studied in details. The hazard rate function of the GPS distribution can be increasing, decreasing, and bathtub-shaped. We obtain several properties of the GPS distribution such as its probability density function, and failure rate function, Shannon entropy, mean residual life function, quantiles, and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented, and simulation studies are performed for evaluation of this estimation for complete data, and the MLE of parameters for censored data. At the end, a real example is given.     

A new absolute continuous bivariate generalized exponential distribution

The generalized exponential is the most commonly used distribution for analyzing lifetime data. This distribution has several desirable properties and it can be used quite effectively to analyse several skewed life time data. The main aim of this paper is to introduce absolutely continuous bivariate generalized exponential distribution using the method of Block and Basu (1974). In fact, the Block and Basu exponential distribution will be extended to the generalized exponential distribution. We call the new proposed model as the Block and Basu bivariate generalized exponential distribution, then, discuss its different properties. In this case the joint probability distribution function and the joint cumulative distribution function can be expressed in compact forms. The model has four unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. To compute the maximum likelihood estimators directly, one needs to solve a four dimensional optimization problem. The EM algorithm has been proposed to compute the maximum likelihood estimations of the unknown parameters. One data analysis is provided for illustrative purposes. Finally, we propose some generalizations of the proposed model and compare their models with each other.