Stochastic comparisons between extreme order statistics from scale models

Stochastic ordering relations between extreme order statistics from exponential, Weibull and gamma distributions have been studied extensively by many researchers in recent years. In this work, we obtain various ordering results for the comparisons of two extreme order statistics from scale models when one set of scale parameters majorizes the other. The new results obtained here are applied when the baseline distributions are exponentiated Weibull or generalized gamma distributions. In this way, we generalize and extend some results established recently in the literature.     

On finite mixtures of geometric and negative binomial distributions

Finite mixtures of distributions have been getting increasing use in the applied literature. In the continuous case, linear combinations of exponentials and gammas have been shown to be well suited for modeling purposes. In the discrete case, the focus has primarily been on continuous mixing, usually of Poisson distributions and typically using gammas to describe the random parameter, But many of these applications are forced, especially when a continuous mixing distribution is used. Instead, it is often prefe-rable to try finite mixtures of geometries or negative binomials, since these are the fundamental building blocks of all discrete random variables. To date, a major stumbling block to their use has been the lack of easy routines for estimating the parameters of such models. This problem has now been alleviated by the adaptation to the discrete case of numerical procedures recently developed for exponential, Weibull, and gamma mixtures. The new methods have been applied to four previously studied data sets, and significant improvements reported in goodness-of-fit, with resultant implications for each affected study.     

A new class of weighted exponential distributions

Introducing a shape parameter to an exponential model is nothing new. There are many ways to introduce a shape parameter to an exponential distribution. The different methods may result in variety of weighted exponential (WE) distributions. In this article, we have introduced a shape parameter to an exponential model using the idea of Azzalini, which results in a new class of WE distributions. This new WE model has the probability density function (PDF) whose shape is very close to the shape of the PDFS of Weibull, gamma or generalized exponential distributions. Therefore, this model can be used as an alternative to any of these distributions. It is observed that this model can also be obtained as a hidden truncation model. Different properties of this new model have been discussed and compared with the corresponding properties of well-known distributions. Two data sets have been analysed for illustrative purposes and it is observed that in both the cases it fits better than Weibull, gamma or generalized exponential distributions.     

Properties of Coherent Systems with Dependent Components

We introduce two new families of univariate distributions that we call hyperminimal and hypermaximal distributions. These families have interesting applications in the context of reliability theory in that they contain that of coherent system lifetime distributions. For these families, we obtain distributions, bounds, and moments. We also define the minimal and maximal signatures of a coherent system with exchangeable components which allow us to represent the system distribution as generalized mixtures (i.e., mixtures with possibly negative weights) of series and parallel systems. These results can also be applied to order statistics (k-out-of-n systems). Finally, we give some applications studying coherent systems with different multivariate exponential joint distributions.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

Generalized exponential distributions

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma 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.     

RELIABILITY PROPERTIES OF SERIES AND PARALLEL SYSTEMS FROM BIVARIATE EXPONENTIAL MODELS

ABSTRACT

In this paper we study the classification of the generalized mixtures of two or three exponential distributions, in the ILR and DLR classes, and consequently in the IFR and DFR classes. We apply these results to classify the aging of the series and parallel systems, in accord with some common bivariate exponential models of their components.     

Distributions of the lifetimes of system components operating under an unknown common environment

SUMMARY Families of joint distributions for describing the lifetimes of a system of components that operate under an unknown environment, when the environment follows a Weibull distribution, are derived. The reliability function for this system is calculated and several properties of the aforementioned joint distributions are investigated.     

Compounded inverse Weibull distributions: Properties,inference and applications

ABSTRACT

In this paper two probability distributions are analyzed which are formed by compounding inverse Weibull with zero-truncated Poisson and geometric distributions. The distributions can be used to model lifetime of series system where the lifetimes follow inverse Weibull distribution and the subgroup size being random follows either geometric or zero-truncated Poisson distribution. Some of the important statistical and reliability properties of each of the distributions are derived. The distributions are found to exhibit both monotone and non-monotone failure rates. The parameters of the distributions are estimated using the expectation-maximization algorithm and the method of minimum distance estimation. The potentials of the distributions are explored through three real life data sets and are compared with similar compounded distributions, viz. Weibull-geometric, Weibull-Poisson, exponential-geometric and exponential-Poisson distributions.     

Moments of order statistics from doubly truncated continuous distributions and characterizations

In this paper, recurrence relations from a general class of doubly truncated continuous distributions which are satisfied by single as well as product moments of order statistics are obtained. Recurrence relations from doubly truncated generalized Weibull, exponential, Raleigh and logistic distributions have been derived as special cases of our result, Some previous results for doubly truncated Weibull, standard exponential, power function and Burr type XII distributions are obtained as special cases. The general recurrence relation of single moments has been used in the case of the left and right truncation to characterize the Weibull, Burr type XII and Pareto distributions.     

Generalized exponential distribution: Existing results and some recent developments

Mudholkar and Srivastava [1993. Exponentiated Weibull family for analyzing bathtub failure data. IEEE Trans. Reliability 42, 299–302] introduced three-parameter exponentiated Weibull distribution. Two-parameter exponentiated exponential or generalized exponential distribution is a particular member of the exponentiated Weibull distribution. Generalized exponential distribution has a right skewed unimodal density function and monotone hazard function similar to the density functions and hazard functions of the gamma and Weibull distributions. It is observed that it can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. The genesis of this model, several properties, different estimation procedures and their properties, estimation of the stress-strength parameter, closeness of this distribution to some of the well-known distribution functions are discussed in this article.     

Empirical approximations for Hoeffding’s test of bivariate independence using two Weibull extensions

The sampling distributions are generally unavailable in exact form and are approximated either in terms of the asymptotic distributions, or their correction using expansions such as Edgeworth, Laguerre or Cornish–Fisher; or by using transformations analogous to that of Wilson and Hilferty. However, when theoretical routes are intractable, in this electronic age, the sampling distributions can be reasonably approximated using empirical methods. The point is illustrated using the null distribution of Hoeffding’s test of bivariate independence which is important because of its consistency against all dependence alternatives. For constructing the approximations we employ two Weibull extensions, the generalized Weibull and the exponentiated Weibull families, which contain a rich variety of density shapes and tail lengths, and have their distribution functions and quantile functions available in closed form, making them convenient for obtaining the necessary percentiles and p-values. Both approximations are seen to be excellent in terms of accuracy, but that based on the generalized Weibull is more portable.     

Estimation of HIV seroconversion and effects of age in the San Francisco homosexual population

SUMMARY Using San Francisco city clinic cohort data, we estimate the HIV seroconversion distribution by both non-parametric and parametric methods, and illustrate the effects of age on this distribution. The non-parametric methods include the Turnbull method, the Bacchetti method, the expectation, maximization and smoothing (EMS) method and the penalized spline method. The seroconversion density curves estimated by these nonparametric methods are of bimodal nature with obvious effects of age. As a result of the bimodal nature of the seroconversion curves, the parametric models considered are mixtures of two distributions taken from the generalized log-logistic distribution with three parameters, the Weibull distribution and the log-normal distribution. In terms of the logarithm of the likelihood values, it appears that the non-parametric methods with smoothing as well as without smoothing (i.e. the Turnbull method) provided much better fits than did the parametric models. Among the non-parametric methods, the EMS and the spline estimates are more appealing, because the unsmoothed Turnbull estimates are very unstable and because the Bacchetti estimates have a longer tail. Among the parametric models, the mixture of a generalized log-logistic distribution with three parameters and a Weibull distribution or a log-normal distribution provided better fits than did other mixtures of parametric models.     

The exponentiated transmuted Weibull geometric distribution with application in survival analysis

This article introduces a new generalization of the transmuted Weibull distribution introduced by Aryal and Tsokos in 2011. We refer to the new distribution as exponentiated transmuted Weibull geometric (ETWG) distribution. The new model contains 22 lifetime distributions as special cases such as the exponentiated Weibull geometric, complementary Weibull geometric, exponentiated transmuted Weibull, exponentiated Weibull, and Weibull distributions, among others. The properties of the new model are discussed and the maximum likelihood estimation is used to evaluate the parameters. Explicit expressions are derived for the moments and examine the order statistics. To examine the performance of our new model in fitting several data we use two real sets of data, censored and uncensored, and then compare the fitting of the new model with some nested and nonnested models, which provides the best fit to all of the data. A simulation has been performed to assess the behavior of the maximum likelihood estimates of the parameters under the finite samples. This model is capable of modeling various shapes of aging and failure criteria.     

Absolute continuous bivariate generalized exponential distribution

Generalized exponential distribution has been used quite effectively to model positively skewed lifetime data as an alternative to the well known Weibull or gamma distributions. In this paper we introduce an absolute continuous bivariate generalized exponential distribution by using a simple transformation from a well known bivariate exchangeable distribution. The marginal distributions of the proposed bivariate generalized exponential distributions are generalized exponential distributions. The joint probability density function and the joint cumulative distribution function can be expressed in closed forms. It is observed that the proposed bivariate distribution can be obtained using Clayton copula with generalized exponential distribution as marginals. We derive different properties of this new distribution. It is a five-parameter distribution, and the maximum likelihood estimators of the unknown parameters cannot be obtained in closed forms. We propose some alternative estimators, which can be obtained quite easily, and they can be used as initial guesses to compute the maximum likelihood estimates. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.     

A study of moments and likelihood estimators of the odd Weibull distribution

The odd Weibull distribution is a three-parameter generalization of the Weibull and the inverse Weibull distributions having rich density and hazard shapes for modeling lifetime data. This paper explored the odd Weibull parameter regions having finite moments and examined the relation to some well-known distributions based on skewness and kurtosis functions. The existence of maximum likelihood estimators have shown with complete data for any sample size. The proof for the uniqueness of these estimators is given only when the absolute value of the second shape parameter is between zero and one. Furthermore, elements of the Fisher information matrix are obtained based on complete data using a single integral representation which have shown to exist for any parameter values. The performance of the odd Weibull distribution over various density and hazard shapes is compared with generalized gamma distribution using two different test statistics. Finally, analysis of two data sets has been performed for illustrative purposes.     

On the choice of precedence tests

The best precedence test (BPT) is derived for testing the hypothesis that the lifetimes of two types of items on test have the same distribution. The test has maximum power in the class of the Lehmann type of alternatives F - 1 - (1-G) , A > 1, where F and G are probability distributions of the lifetimes of two types of items on test. This class includes exponential distributions, the Weibull distribution differing only in scale and distributions with proportional hazard rates. Exact power of the BPT is compared with other nonparametrie and parametric tests. The test may terminate before all the lifetimes of the items on test are recorded. In comparing with competing tests of equal size, the power functions are similar but a considerable number of items can be saved and the time on test can be reduced by using the BPT     

The Kumaraswamy-transmuted exponentiated modified Weibull distribution

This article introduces a new generalization of the transmuted exponentiated modified Weibull distribution introduced by Eltehiwy and Ashour in 2013, using Kumaraswamy distribution introduced by Cordeiro and de Castro in 2011. We refer to the new distribution as Kumaraswamy-transmuted exponentiated modified Weibull (Kw-TEMW) distribution. The new model contains 54 lifetime distributions as special cases such as the KumaraswamyWeibull, exponentiated modified Weibull, exponentiated Weibull, exponentiated exponential, transmuted Weibull, Rayleigh, linear failure rate, and exponential distributions, among others. The properties of the new model are discussed and the maximum likelihood estimation is used to evaluate the parameters. Explicit expressions are derived for the moments and examine the order statistics. This model is capable of modeling various shapes of aging and failure criteria.     

Comparisons between parallel systems with exponentiated generalized gamma components

Consider two parallel systems with their independent components’ lifetimes following heterogeneous exponentiated generalized gamma distributions, where the heterogeneity is in both shape and scale parameters. We then obtain the usual stochastic (reversed hazard rate) order between the lifetimes of two systems by using the weak submajorization order between the vectors of shape parameters and the p-larger (weak supermajorization) order between the vectors of scale parameters, under some restrictions on the involved parameters. Further, by reducing the heterogeneity of parameters in each system, the usual stochastic (reversed hazard rate) order mentioned above is strengthened to the hazard rate (likelihood ratio) order. Finally, two characterization results concerning the comparisons of two parallel systems, one with independent heterogeneous generalized exponential components and another with independent homogeneous generalized exponential components, are derived. These characterization results enable us to find some lower and upper bounds for the hazard rate and reversed hazard rate functions of a parallel system consisting of independent heterogeneous generalized exponential components. The results established here generalize some of the known results in the literature, concerning the comparisons of parallel systems under generalized exponential and exponentiated Weibull models.