A new four-parameter discrete distribution with bathtub and unimodal failure rate

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.     

A new generalization of the Weibull-geometric distribution with bathtub failure rate

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.     

The TTT transformation and a new bathtub distribution model

In this paper we use the total time on test transformation to establish a method for construction of parametric models of lifetime distributions having bathtub-shaped failure rate. We study a particular model which is simple compared to the other existing models. We derive expressions for moments and quantiles and treat estimation methods. Particularly, the maximum likelihood method is studied. Consistency proofs are given.     

Change-point analysis with bathtub shape for the exponential distribution

Likelihood ratio type test statistic and Schwarz information criterion statistics are proposed for detecting possible bathtub-shaped changes in the parameter in a sequence of exponential distributions. The asymptotic distribution of likelihood ratio type statistic under the null hypothesis and the testing procedure based on Schwarz information criterion are derived. Numerical critical values and powers of two methods are tabulated for certain selected values of the parameters. The tests are applied to detect the change points for the predator data and Stanford heart transplant data.     

Life expectancy of the population in Silesia

The author analyzes changes in average life expectancy in Silesian voivodships in Poland from 1976 to 1995. Differences according to sex, age, and place of residence are considered. (ANNOTATION)     

Transformation of the bathtub failure rate data in reliability for using Weibull-model analysis

All statistical methods involve basic model assumptions, which if violated render results of the analysis dubious. A solution to such a contingency is to seek an appropriate model or to modify the customary model by introducing additional parameters. Both of these approaches are in general cumbersome and demand uncommon expertise. An alternative is to transform the data to achieve compatibility with a well understood and convenient customary model with readily available software. The well-known example is the Box–Cox data transformation developed in order to make the normal theory linear model usable even when the assumptions of normality and homoscedasticity are not met.In reliability analysis the model appropriateness is determined by the nature of the hazard function. The well-known Weibull distribution is the most commonly employed model for this purpose. However, this model, which allows only a small spectrum of monotone hazard rates, is especially inappropriate if the data indicate bathtub-shaped hazard rates.In this paper, a new model based on the use of data transformation is presented for modeling bathtub-shaped hazard rates. Parameter estimation methods are studied for this new (transformation) approach. Examples and results of comparisons between the new model and other bathtub-shaped models are shown to illustrate the applicability of this new model.     

Exponentiated Teissier distribution with increasing,decreasing and bathtub hazard functions

This article introduces a two-parameter exponentiated Teissier distribution. It is the main advantage of the distribution to have increasing, decreasing and bathtub shapes for its hazard rate function. The expressions of the ordinary moments, identifiability, quantiles, moments of order statistics, mean residual life function and entropy measure are derived. The skewness and kurtosis of the distribution are explored using the quantiles. In order to study two independent random variables, stress–strength reliability and stochastic orderings are discussed. Estimators based on likelihood, least squares, weighted least squares and product spacings are constructed for estimating the unknown parameters of the distribution. An algorithm is presented for random sample generation from the distribution. Simulation experiments are conducted to compare the performances of the considered estimators of the parameters and percentiles. Three sets of real data are fitted by using the proposed distribution over the competing distributions.     

Estimation and prediction for Chen distribution with bathtub shape under progressive censoring

We consider estimation of the unknown parameters of Chen distribution [Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statist Probab Lett. 2000;49:155–161] with bathtub shape using progressive-censored samples. We obtain maximum likelihood estimates by making use of an expectation–maximization algorithm. Different Bayes estimates are derived under squared error and balanced squared error loss functions. It is observed that the associated posterior distribution appears in an intractable form. So we have used an approximation method to compute these estimates. A Metropolis–Hasting algorithm is also proposed and some more approximate Bayes estimates are obtained. Asymptotic confidence interval is constructed using observed Fisher information matrix. Bootstrap intervals are proposed as well. Sample generated from MH algorithm are further used in the construction of HPD intervals. Finally, we have obtained prediction intervals and estimates for future observations in one- and two-sample situations. A numerical study is conducted to compare the performance of proposed methods using simulations. Finally, we analyse real data sets for illustration purposes.     

The exact hypothesis test for the shape parameter of a new two-parameter distribution with the bathtub shape or increasing failure rate function under progressive censoring with random removals

This study considers the exact hypothesis test for the shape parameter of a new two-parameter distribution with the shape of a bathtub or increasing failure rate function under type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial or a uniform distribution. Several test statistics are proposed and one numerical example is provided to illustrate the proposed hypothesis test for the shape parameter. Finally, a simulation study is performed to compare the power performances of all proposed test statistics. We concluded that the test statistic w ₁ is more attractive than other methods as it has better performance than other test statistics for most cases based on the criteria of maximum power.     

The generalized inverse Lindley distribution: A new inverse statistical model for the study of upside-down bathtub data

ABSTRACT

In this article, a two-parameter generalized inverse Lindley distribution capable of modeling a upside-down bathtub-shaped hazard rate function is introduced. Some statistical properties of proposed distribution are explicitly derived here. The method of maximum likelihood, least square, and maximum product spacings are used for estimating the unknown model parameters and also compared through the simulation study. The approximate confidence intervals, based on a normal and a log-normal approximation, are also computed. Two algorithms are proposed for generating a random sample from the proposed distribution. A real data set is modeled to illustrate its applicability, and it is shown that our distribution fits much better than some other existing inverse distributions.     

Estimating the turning point of a bathtub-shaped failure distribution

The turning point of a hazard rate function is useful in assessing the hazard in the useful life phase and helps to determine and plan appropriate burn-in, maintenance, and repair policies and strategies. For many bathtub-shaped distributions, the turning point is unique, and the hazard varies little in the useful life phase. We investigate the performance of an empirical estimator for the turning point in the case of the modified Weibull distribution, a bathtub-shaped generalization of the Weibull distribution, that has been found to be useful in reliability engineering and other areas concerned with life-time data. We illustrate the theory by means of an example, and also conduct a simulation study to assess the performance of the estimator in practice.     

A new generalized weighted Weibull distribution with decreasing,increasing, upside-down bathtub,N-shape and M-shape hazard rate

Recently, Domma et al. [An extension of Azzalinis method, J. Comput. Appl. Math. 278 (2015), pp. 37–47] proposed an extension of Azzalini's method. This method can attract readers due to its flexibility and ease of applicability. Most of the weighted Weibull models that have been introduced are with monotonic hazard rate function. This fact limits their applicability. So, our aim is to build a new weighted Weibull distribution with monotonic and non-monotonic hazard rate function. A new weighted Weibull distribution, so-called generalized weighted Weibull (GWW) distribution, is introduced by a method exposed in Domma et al. [13]. GWW distribution possesses decreasing, increasing, upside-down bathtub, N-shape and M-shape hazard rate. Also, it is very easy to derive statistical properties of the GWW distribution. Finally, we consider application of the GWW model on a real data set, providing simulation study too.     

Estimation of parameters and the mean life of a mixed failure time distribution

This paper deals with estimation of parameters and the mean life of a mixed failure time distribution that has a discrete probability mass at zero and an exponential distribution with mean O for positive values. A new sampling scheme similar to Jayade and Prasad (1990) is proposed for estimation of parameters. We derive expressions for biases and mean square errors (MSEs) of the maximum likelihood estimators (MLEs). We also obtain the uniformly minimum variance unbiased estimators (UMVUEs) of the parameters. We compare the estimator of O and mean life fj based on the proposed sampling scheme with the estimators obtained by using the sampling scheme of Jayade and Prasad (1990).     

Estimation of parameters of mixed failure time distribution

The problem of estimation of parameters of a mixture of degenerate and exponential distributions is considered. A new sampling scheme is proposed and the exact bias and the mean square error (MSE) of the maximum likelihood estimators of the parameters is derived. Moment estimators, their approximate biases and the MSE are obtained. Asymptotic distributions of the estimators are also obtained for both the cases.     

Hybrid censoring schemes with exponential failure distribution

The mixture of Type I and Type I1 censoring schemes, called the hybrid censoring, is quite important in life–testing experiments. Epstein(1954, 1960) introduced this testing scheme and proposed a two–sided confidence interval to estimate the mean lifetime, θ, when the underlying lifetime distribution is assumed to be exponential. There are some two–sided confidence intervals and credible intervals proposed by Fairbanks et al. (1982) and Draper and Guttman (1987) respectively. In this paper we obtain the exact two–sided confidence interval of θ following the approach of Chen and Bhattacharya (1988). We also obtain the asymptotic confidence intervals in the Hybrid censoring case. It is important to observe that the results for Type I and Type II censoring schemes can be obtained as particular cases of the Hybrid censoring scheme. We analyze one data set and compare different methods by Monte Carlo simulations.     

Generalized linear failure rate power series distribution

Abstract

In this article, we introduce a new class of lifetime distributions. This new class includes several previously known distributions such as those of Chahkandi and Ganjali (2009 Chahkandi, M., Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computat. Statist. Data Anal. 53:4433–4440.[Crossref], [Web of Science ®] , [Google Scholar]), Mahmoudi and Jafari (2012 Mahmoudi, E., Jafari, A.A. (2012). Generalized exponential power series distributions. Comput. Statist. Data Anal. 56(12):4047–4066.[Crossref], [Web of Science ®] , [Google Scholar]), and Nadarajah et al. (2012 Nadarajah, S., Shahsanaei, F., Rezaei, S. (2012). A new four-parameter lifetime distribution. J. Statist. Computat. Simul.. ifirst, 1–16. [Google Scholar]). This new class of four-parameter distributions allows for flexible failure rate behavior. Indeed, the failure rate function here can be increasing, decreasing, bathtub-shaped or upside-down bathtub-shaped. Several distributional properties of the new class including moments, quantiles and order statistics are studied. An EM algorithm for computing the estimates of the parameters involved is proposed and some maximum entropy characterizations are discussed. Finally, to show the flexibility and potential of the new class of distributions, applications to two real data sets are provided.     

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.     

Analysis op failure time data by burr distribution

Sometimes it is appropriate to model the survival and failure time data by a non-monotonic failure rate distribution. This may be desirable when the course of disease is such that mortality reaches a peak after some finite period and then slowly declines.In this paper we study Burr, type XII model whose failure rate exhibits the above behavior. The location of the critical points (at which the monotonicity changes) for both the failure rate and the mean residual life function (MRLF) are studied. A procedure is described for estimating these critical points. Necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimates are provided and it is shown that the conditions provided by Wingo (1993) are not sufficient. A data set pertaining to fibre failure strengths is analyzed and the maximum likelihood estimates of the critical points are obtained.