Reliability study of proportional odds family of discrete distributions

The proportional odds model gives a method of generating new family of distributions by adding a parameter, called tilt parameter, to expand an existing family of distributions. The new family of distributions so obtained is known as Marshall–Olkin family of distributions or Marshall–Olkin extended distributions. In this paper, we consider Marshall–Olkin family of distributions in discrete case with fixed tilt parameter. We study different ageing properties, as well as different stochastic orderings of this family of distributions. All the results of this paper are supported by several examples.     

A discrete analogue of odd Weibull-G family of distributions: properties,classical and Bayesian estimation with applications to count data

In the statistical literature, several discrete distributions have been developed so far. However, in this progressive technological era, the data generated from different fields is getting complicated day by day, making it difficult to analyze this real data through the various discrete distributions available in the existing literature. In this context, we have proposed a new flexible family of discrete models named discrete odd Weibull-G (DOW-G) family. Its several impressive distributional characteristics are derived. A key feature of the proposed family is its failure rate function that can take a variety of shapes for distinct values of the unknown parameters, like decreasing, increasing, constant, J-, and bathtub-shaped. Furthermore, the presented family not only adequately captures the skewed and symmetric data sets, but it can also provide a better fit to equi-, over-, under-dispersed data. After producing the general class, two particular distributions of the DOW-G family are extensively studied. The parameters estimation of the proposed family, are explored by the method of maximum likelihood and Bayesian approach. A compact Monte Carlo simulation study is performed to assess the behavior of the estimation methods. Finally, we have explained the usefulness of the proposed family by using two different real data sets.     

Some bounds related to Harris family of distributions

As a lifetime distribution, Harris family of distributions are applied to the lifetime of a series system with random number of components. In this paper, properties of various ageing classes of mixtures of Harris family of distributions, where the tilt parameter of a Harris distribution is taken as a random variable, are studied. We obtain an upper bound for maximum error in evaluating its reliability function. Two bounds are also presented for survival function and expectation of the mixed Harris family. We also provide some interesting bounds for its residual survival function. Our results generalize several previous findings in this connection. Some illustrative examples are also provided.     

Discrete distributions in the extended FGM family: The p.g.f. approach

In this article the probability generating functions of the extended Farlie–Gumbel–Morgenstern family for discrete distributions are derived. Using the probability generating function approach various properties are examined, the expressions for probabilities, moments, and the form of the conditional distributions are obtained. Bivariate version of the geometric and Poisson distributions are used as illustrative examples. Their covariance structure and estimation of parameters for a data set are briefly discussed. A new copula is also introduced.     

On an Extension of the Exponentiated Weibull Distribution

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.     

A new generalized odd log-logistic family of distributions

We introduce and study general mathematical properties of a new generator of continuous distributions with three extra parameters called the new generalized odd log-logistic family of distributions. The proposed family contains several important classes discussed in the literature as submodels such as the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, entropy measures, and order statistics, which hold for any baseline model, are presented. We also present certain characterization of the proposed distribution and derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behavior of the maximum likelihood estimator via simulation. The importance of the new family is illustrated by means of two real data sets. These applications indicate that the new family can provide better fits than other well-known classes of distributions. The beauty and importance of the new family lies in its ability to model real data.     

Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

The Harris Extended Exponential Distribution

A new lifetime distribution is proposed and studied. The Harris extended exponential is obtained from a mixture of the exponential and Harris distributions, which arises from a branching process. Several structural properties of the new distribution are discussed, including moments, generating function and order statistics. The new distribution can model data with increasing or decreasing failure rate. The shape of the hazard rate function is controlled by one of the added parameters in an uncomplicated manner. An application to a real dataset illustrates the usefulness of the new distribution.     

A generalization of discrete Weibull distribution

In this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.     

On families of beta- and generalized gamma-generated distributions and associated inference

A general family of univariate distributions generated by beta random variables, proposed by Jones, has been discussed recently in the literature. This family of distributions possesses great flexibility while fitting symmetric as well as skewed models with varying tail weights. In a similar vein, we define here a family of univariate distributions generated by Stacy’s generalized gamma variables. For these two families of univariate distributions, we discuss maximum entropy characterizations under suitable constraints. Based on these characterizations, an expected ratio of quantile densities is proposed for the discrimination of members of these two broad families of distributions. Several special cases of these results are then highlighted. An alternative to the usual method of moments is also proposed for the estimation of the parameters, and the form of these estimators is particularly amenable to these two families of distributions.     

On the discrete analogues of continuous distributions

In this paper, a new method is proposed for generating discrete distributions. A special class of the distributions, namely, the T-geometric family contains the discrete analogues of continuous distributions. Some general properties of the T-geometric family of distributions are obtained. A member of the T-geometric family, namely, the exponentiated-exponential–geometric distribution is defined and studied. Various properties of the exponentiated-exponential–geometric distribution such as the unimodality, the moments and the probability generating function are discussed. The method of maximum likelihood estimation is proposed for estimating the model parameters. Three real data sets are used to illustrate the applications of the exponentiated-exponential–geometric distribution.     

On the discrete analogues of continuous distributions

In this paper, a new method is proposed for generating discrete distributions. A special class of the distributions, namely, the T-geometric family contains the discrete analogues of continuous distributions. Some general properties of the T-geometric family of distributions are obtained. A member of the T-geometric family, namely, the exponentiated-exponential–geometric distribution is defined and studied. Various properties of the exponentiated-exponential–geometric distribution such as the unimodality, the moments and the probability generating function are discussed. The method of maximum likelihood estimation is proposed for estimating the model parameters. Three real data sets are used to illustrate the applications of the exponentiated-exponential–geometric distribution.     

An Extension of the Normal/Independent Family for Studies of Robust Estimators

The family of normal scale mixture distributions, also called the Normal/Independent family, has been used for efficient Monte Carlo studies of robust estimators. The distributions in this family are unimodal. The Normal/Independent family is extended by introducing a location mixing in addition to the scale mixing. Distributions in this extension may be nonunimodal. The asymptotic variances of robust estimators of location are compared using the distributions from the extension. A Monte Carlo swindle similar to the one used in the Princeton study is given for the extended family. A small simulation study demonstrates the efficiency of the swindle. The swindle is compared with other swindle methods based on Fisher's score function and regression.     

A bayesian approach to prediction in the presence of spurious observations for several models

The problem of spuriousity has been dealt with from a Bayesian perspective by, among others, Box and Taio (1968) and in several papers by Guttman with various co-authors, beginning with Guttman (1973), The main objective of these papers has been to obtain posterior distributions of parameters, and to base inference on these distributions. In the current paper, the Bayesian argument is carried one step further by deriving predictive distributions of future observations. Inferences are then based on these distributions. We will obtain predictive results for several models, First, we consider the univariate normal case with one spurious observation, This is then generalized to several spurious observations. The multivariate normal situation is studied next. Finally, we consider the general linear model with normal errors.     

Survival distributions arising from two families and generated by transformations

Two families of distributions are introduced and studied within the framework of parametric survival analysis. The families are derived from a general linear form by specifying a function of the survival function with certain restrictions. Distributions within each family are generated by transformations of the survival time variable subject to certain restrictions. Two specific transformations were selected and, thus, four distributions are identified for further study. The distributions have one scale and two shape parameters and include as special cases the exponential, Weibull, log-logistic and Gompertz distributions. One of the new distributions, the modified Weibull, is studied in some detail.

The distributions are developed with an emphasis on those features that data analysts find especially useful for survivorship studies, A wide variety of hazard shapes are available. The survival, density and hazard functions may be written in simple algebraic forms. Parameter estimation is demonstrated using the least squares and maximum likelihood methods. Graphical techniques to assess goodness of fit are demonstrated. The models may be extended to include concmitant information.     

General properties for the beta extended half-normal model

We formulate and study a four-parameter lifetime model called the beta extended half-normal distribution. This model includes as sub-models the exponential, extended half-normal and half-normal distributions. We derive expansions for the new density function which do not depend on complicated functions. We obtain explicit expressions for the moments and incomplete moments, generating function, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. In addition, the model parameters are estimated by maximum likelihood. We provide the observed information matrix. The new model is modified to cope with possible long-term survivors in the data. The usefulness of the new distribution is shown by means of two real data sets.     

Odd-Burr generalized family of distributions with some applications

We study a new family of continuous distributions with two extra shape parameters called the Burr generalized family of distributions. We investigate the shapes of the density and hazard rate function. We derive explicit expressions for some of its mathematical quantities. The estimation of the model parameters is performed by maximum likelihood. We prove the flexibility of the new family by means of applications to two real data sets. Furthermore, we propose a new extended regression model based on the logarithm of the Burr generalized distribution. This model can be very useful to the analysis of real data and provide more realistic fits than other special regression models.     

Model Selection Using Conditional Densities

In this article we propose a new method to select a discrete model f(x; θ), based on the conditional density of a sample given the value of a sufficient statistic for θ. The main idea is to work with a broad family of discrete distributions, called the family of power series distribution, for which there is a common sufficient statistic for the parameter of interest. The proposed method uses the maximum conditional density in order to select the best model.

We compare our proposal with the usual methodology based on Bayes factors. We provide several examples that show that our proposal works fine in most instances. Bayes factors are strongly dependent on the prior information about the parameters. Since our method does not require the specification of a prior distribution, it provides a useful alternative to Bayes factors.     

The exponentiated transmuted-G family of distributions: Theory and applications

In this paper, a new family of continuous distributions called the exponentiated transmuted-G family is proposed which extends the transmuted-G family defined by Shaw and Buckley (2007). Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, and order statistics are derived. Some special models of the new family are provided. The maximum likelihood is used for estimating the model parameters. We provide the simulation results to assess the performance of the proposed model. The usefulness and flexibility of the new family is illustrated using real data.     

Statistical analysis of the Lomax–Logarithmic distribution

In this paper we introduce a three-parameter lifetime distribution following the Marshall and Olkin [New method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 1997;84(3):641–652] approach. The proposed distribution is a compound of the Lomax and Logarithmic distributions (LLD). We provide a comprehensive study of the mathematical properties of the LLD. In particular, the density function, the shape of the hazard rate function, a general expansion for moments, the density of the rth order statistics, and the mean and median deviations of the LLD are derived and studied in detail. The maximum likelihood estimators of the three unknown parameters of LLD are obtained. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance–covariance matrix. Finally, a real data set is analysed to show the potential of the new proposed distribution.