Optimal Multiple Constant-Stress Accelerated Life Tests for Generalized Exponential Distribution

Recently generalized exponential distribution has been discussed by many authors. In this article, we study the optimal constant-stress accelerated life tests with complete sample for the generalized exponential distribution. The problem of choosing the optimal proportions of test units allocated to each stress level is addressed by using V-optimality as well as D-optimality criteria. Some interesting conclusions are obtained. Finally, real data example and numerical examples have been analyzed to illustrate the proposed procedures.     

Generalized Weighted Exponential Distribution

The new class of weighted exponential (WE) distributions obtained by Gupta and Kundu (2009) by implementing Azzalini's method to the exponential distribution. In this study, we generalize the WE distribution to a new class of generalized weighted exponential (GWE) distribution. Several statistical and reliability properties of this new class of distribution are obtained. Estimation and inference procedure for distribution parameters are investigated. Finally, we show that the proposed model can provide better fit than the recent class of weighted exponential by using two real data examples.     

Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

Statistical Analysis in Constant-stress Accelerated Life Tests for Generalized Exponential Distribution with Progressive Type-I Hybrid Censoring

Based on progressive Type-I hybrid censored data, statistical analysis in constant-stress accelerated life test (CS-ALT) for generalized exponential (GE) distribution is discussed. The maximum likelihood estimates (MLEs) of the parameters and the reliability function are obtained with EM algorithm, as well as the observed Fisher information matrix, the asymptotic variance-covariance matrix of the MLEs, and the asymptotic unbiased estimate (AUE) of the scale parameter. Confidence intervals (CIs) for the parameters are derived using asymptotic normality of MLEs and percentile bootstrap (Boot-p) method. Finally, the point estimates and interval estimates of the parameters are compared separately through the Monte-Carlo method.     

混合指数几何分布

本文讨论元件的寿命分布服从双参数混合指数分布,元件个数服从几何分布的情形下,定义了混合指数几何分布,并研究了该分布的各种性质,讨论了参数的极大似然估计,并使用EM算法得到参数的近似估计. 最后,本文给出了参数的渐近方差、协方差、置信区间.     

Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Distribution

In this article, a transmuted linear exponential distribution is developed that generalizes the linear exponential distribution with an additional parameter using the quadratic rank transmutation map which was studied by Shaw et al. Some statistical properties of the proposed distribution such as moments, quantiles, and the failure rate function are investigated. The maximum likelihood estimators of unknown parameters are also discussed and a real data analysis is carried out to illustrate the superiority of the proposed distribution.     

Test of Normality Against Generalized Exponential Power Alternatives

The family of symmetric generalized exponential power (GEP) densities offers a wide range of tail behaviors, which may be exponential, polynomial, and/or logarithmic. In this article, a test of normality based on Rao's score statistic and this family of GEP alternatives is proposed. This test is tailored to detect departures from normality in the tails of the distribution. The main interest of this approach is that it provides a test with a large family of symmetric alternatives having non-normal tails. In addition, the test's statistic consists of a combination of three quantities that can be interpreted as new measures of tail thickness. In a Monte-Carlo simulation study, the proposed test is shown to perform well in terms of power when compared to its competitors.     

A Generalization of the Univariate Slash by a Scale-Mixtured Exponential Power Distribution

A generalization of the slash distribution is derived using the scale mixture of the exponential power distribution. The newly defined family of distributions provides a rich flexibility on the tail heaviness and yields alternative robust estimators of location and scale in non normal situations. In order to investigate asymptotically the bias properties of the estimators, a simulation study is performed. The performance of the estimators on two well-known real data sets is also illustrated.     

Bayes Inference in Constant Partially Accelerated Life Tests for the Generalized Exponential Distribution with Progressive Censoring

In this paper, the problem of constant partially accelerated life tests when the lifetime follows the generalized exponential distribution is considered. Based on progressive type-II censoring scheme, the maximum likelihood and Bayes methods of estimation are used for estimating the distribution parameters and acceleration factor. A Monte Carlo simulation study is carried out to examine the performance of the obtained estimates.     

A Mixed Bivariate Distribution Connected with Geometric Maxima of Exponential Variables

We study the joint distribution of X and N, where N has a geometric distribution and X is the maximum of N i.i.d. exponential variables, independent of N. We present basic properties of these mixed bivariate distributions and discuss parameter estimation for this model. An example from finance, where N represents the number of consecutive positive daily log-returns of currency exchange rates, illustrates stochastic modeling potential of these laws.     

Discrete Beta-Exponential Distribution

Consider the estimation of the regression parameters in the usual linear model. For design densities with infinite support, it has been shown by Faraldo Roca and González Manteiga [1] Faraldo Roca, P. and González Manteiga, W. 1987. “Efficiency of a new class of linear regression estimates obtained by preliminary nonparametric estimation”. In New Perspectives in Theoretical and Applied Statistics Edited by: Puri, M. L., Vilaplana, J. P. and Wertz, W. 229–242. New York: John Wiley.  [Google Scholar] that it is possible to modify the classical least squares procedure and to obtain estimators for the regression parameters whose MSE's (mean squared errors) are smaller than those of the usual least squares estimators. The modification consists of presmoothing the response variables by a kernel estimator of the regression function. These authors also show that the gain in efficiency is not possible for a design density with compact support. We show that in this case local linear presmoothing does not fix this inefficiency problem, in spite of the well known fact that local linear fitting automatically corrects the bias in the endpoints of the (design density) support. We demonstrate on a theoretical basis how this inefficiency problem can be rectified in the compact design case: we prove that presmoothing with boundary kernels studied in Müller [2] Müller, H.-G. 1991. Smooth optimum kernel estimators near endpoints. Biometrika, 78: 521–530. [Crossref], [Web of Science ®] , [Google Scholar] and Müller and Wang [3] Müller, H.-G. and Wang, J.-L. 1994. Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics, 50: 61–76. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] leads to regression estimators which are superior over the least squares estimators. A very careful analytic treatment is needed to arrive at these asymptotic results.     

A New Generalized Exponential Geometric Distribution

In this article, another version of the generalized exponential geometric distribution different to that of Silva et al. (2010 Silva , R. B. , Barreto-Souza , W. , Cordeiro , G. M. ( 2010 ). A new distribution with decreasing, increasing and upside-down bathtub failure rate. Computat. Statist. Data Anal. 54: 935–944 . [Google Scholar]) is proposed. This new three-parameter lifetime distribution with decreasing, increasing, and bathtub failure rate function is created by compounding the generalized exponential distribution of Gupta and Kundu (1999 Gupta , R. D. , Kundu , D. ( 1999 ). Generalized exponential distributions . Austral. NZ J. Statist. 41 ( 2 ): 173 – 188 .[Crossref], [Web of Science ®] , [Google Scholar]) with a geometric distribution. Some basic distributional properties, moment-generating function, rth moment, and Rényi entropy of the new distribution are studied. The model parameters are estimated by the maximum likelihood method and the asymptotic distribution of estimators is discussed. Finally, an application of the new distribution is illustrated using the two real data sets.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

Exact Inference for a Simple Step-stress Model with Generalized Type-I Hybrid Censored Data from the Exponential Distribution

In this article, the simple step-stress model is considered based on generalized Type-I hybrid censored data from the exponential distribution. The maximum likelihood estimators (MLEs) of the unknown parameters are derived assuming a cumulative exposure model. We then derive the exact distributions of the MLEs of the parameters using conditional moment generating functions. The Bayesian estimators of the parameters are derived and then compared with the MLEs. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs, Bayesian, and the parametric bootstrap methods. The problem of determining the optimal stress-changing point is discussed and the MLEs of the pth quantile and reliability functions at the use condition are obtained. Finally, Monte Carlo simulation and some numerical results are presented for illustrating all the inferential methods developed here.     

Truncated inverted generalized exponential distribution and its properties

The inverted generalized exponential distribution is defined as an alternative model for lifetime data. The existence of moments of this distribution is shown to hold under some restrictions. However, all the moments exist for the truncated inverted generalized exponential distribution and closed-form expressions for them are derived in this article. The distributional properties of this truncated distribution are studied. Maximum likelihood estimation method is discussed for the estimation of the parameters of the distribution both theoretically and empirically. In order to see the modeling performance of the distribution, two real datasets are analyzed.     

Comparison of Shared Frailty Models for Kidney Infection Data under Exponential Power Baseline Distribution

Shared frailty models are often used to model heterogeneity in survival analysis. There are certain assumptions about the baseline distribution and distribution of frailty. In this paper, four shared frailty models with frailty distribution gamma, inverse Gaussian, compound Poisson, and compound negative binomial with exponential power as baseline distribution are proposed. These models are fitted using Markov Chain Monte Carlo methods. These models are illustrated with a real life bivariate survival data set of McGilchrist and Aisbett (1991) related to kidney infection, and the best model is suggested for the data using different model comparison criteria.     

Comparison of Interval Estimators of Pr(X < Y) in the Two-parameter Exponential Distribution

We consider confidence intervals for the stress–strength reliability Pr(X< Y) in the two-parameter exponential distribution. We have derived the Bayesian highest posterior density interval using non-informative prior distributions. We have compared its performance with the intervals based on the generalized pivot variable intervals in terms of their coverage probabilities and expected lengths. Our simulation study shows that the Bayesian interval performs better according to the criteria used, especially when the sample sizes are very small. An example is given.     

The Joint Distribution of the Sum and the Maximum of IID Exponential Random Variables

The authors establish the joint distribution of the sum X and the maximum Y of IID exponential random variables. They derive exact formuli describing the random vector (X, Y), including its joint PDF, CDF, and other characteristics; marginal and conditional distributions; moments and related parameters; and stochastic representations leading to further properties of infinite divisibility and self-decomposability. The authors also discuss parameter estimation and include an example from climatology that illustrates the modeling potential of this new bivariate model.     

Generalized Inverse Gamma Distribution and its Application in Reliability

In this article, we introduce a new reliability model of inverse gamma distribution referred to as the generalized inverse gamma distribution (GIG). A generalization of inverse gamma distribution is defined based on the exact form of generalized gamma function of Kobayashi (1991). This function is useful in many problems of diffraction theory and corrosion problems in new machines. The new distribution has a number of lifetime special sub-models. For this model, some of its statistical properties are studied. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived. We also demonstrate the usefulness of this distribution on a real data set.