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.     

An Efficient Method of Parameter and Quantile Estimation for the Three-Parameter Weibull Distribution Based on Statistics Invariant to Unknown Location Parameter

The three-parameter Weibull distribution is widely used in life testing and reliability analysis. In this article, we propose an efficient method for the estimation of parameters and quantiles of the three-parameter Weibull distribution, which avoids the problem of unbounded likelihood, by using statistics invariant to unknown location. Through a Monte Carlo simulation study, we show that the proposed method performs well compared to other prominent methods based on bias and MSE. Finally, we present two illustrative examples.     

A Finite Mixture Three-Parameter Weibull Model for the Analysis of Wind Speed Data

This article presents a mixture three-parameter Weibull distribution to model wind speed data. The parameters are estimated by using maximum likelihood (ML) method in which the maximization problem is regarded as a nonlinear programming with only inequality constraints and is solved numerically by the interior-point method. By applying this model to four lattice-point wind speed sequences extracted from National Centers for Environmental Prediction (NCEP) reanalysis data, it is observed that the mixture three-parameter Weibull distribution model proposed in this paper provides a better fit than the existing Weibull models for the analysis of wind speed data under study.     

Moment expressions and summary statistics for the complete and truncated weibull distribution

Traditionally, the moments of the Weibull distribution have been calculated using the standard Weibull (Johnson and Kotz, 1970) . This article will expand on that idea and cover the truncated cases for the standard Weibull distributions. Also, the same techniques used for the standard form will be used to derive the moment expressions for the three-parameter complete and truncated Weibull distributions. The summary statistics are then calculated from the moment expressions. Weibull moments involve the gamma and incomplete gamma functions.     

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.     

Step partially accelerated life tests under finite mixture models

In this paper, step partially accelerated life tests are considered when the lifetime of an item under use condition follows a finite mixture of distributions. The analysis is performed when each of the components follows a general class of distributions, which includes, among others, the Weibull, compound Weibull (or three-parameter Burr type XII), power function, Gompertz and compound Gompertz distributions. Based on type-I censoring, the maximum likelihood estimates (MLEs) of the mixing proportions, scale parameters and acceleration factor are obtained. Special attention is paid to a mixture of two exponential components. Simulation results are obtained to study the precision of MLEs.     

The generalized inverse Weibull distribution

The inverse Weibull distribution has the ability to model failure rates which are quite common in reliability and biological studies. A three-parameter generalized inverse Weibull distribution with decreasing and unimodal failure rate is introduced and studied. We provide a comprehensive treatment of the mathematical properties of the new distribution including expressions for the moment generating function and the rth generalized moment. The mixture model of two generalized inverse Weibull distributions is investigated. The identifiability property of the mixture model is demonstrated. For the first time, we propose a location-scale regression model based on the log-generalized inverse Weibull distribution for modeling lifetime data. In addition, we develop some diagnostic tools for sensitivity analysis. Two applications of real data are given to illustrate the potentiality of the proposed regression model.     

Bayesian predictions for compound weibull model

This paper is concerned with a full Bayesian analysis for some prediction problems of the compound model if the underlying distributions are Weibull with unequal shape parameters and the sample is type I censored. A numerical example and computer facilities will be used to illustrate the procedure.     

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.     

Bayesian predictive density of order statistics based on finite mixture models

Bayesian predictive density functions, which are necessary to obtain bounds for predictive intervals of future order statistics, are obtained when the population density is a finite mixture of general components. Such components include, among others, the Weibull (exponential and Rayleigh as special cases), compound Weibull (three-parameter Burr type XII), Pareto, beta, Gompertz and compound Gompertz distributions. The prior belief of the experimenter is measured by a general distribution that was suggested by AL-Hussaini (J. Statist. Plann. Inf. 79 (1999b) 79). Applications to finite mixtures of Weibull and Burr type XII components are illustrated and comparison is made, in the special cases of the exponential and Pareto type II components, with previous results.     

Simplified Likelihood Based Goodness-of-fit Tests for the Weibull Distribution

The aim of this paper is to present new likelihood based goodness-of-fit tests for the two-parameter Weibull distribution. These tests consist in nesting the Weibull distribution in three-parameter generalized Weibull families and testing the value of the third parameter by using the Wald, score, and likelihood ratio procedures. We simplify the usual likelihood based tests by getting rid of the nuisance parameters, using three estimation methods. The proposed tests are not asymptotic. A comprehensive comparison study is presented. Among a large range of possible GOF tests, the best ones are identified. The results depend strongly on the shape of the underlying hazard rate.     

Transmuted Weibull distribution: Properties and estimation

In this article, we investigate the potential usefulness of the three-parameter transmuted Weibull distribution for modeling survival data. The main advantage of this distribution is that it has increasing, decreasing or constant instantaneous failure rate depending on the shape parameter and the new transmuting parameter. We obtain several mathematical properties of the transmuted Weibull distribution such as the expressions for the quantile function, moments, geometric mean, harmonic mean, Shannon, Rényi and q-entropies, mean deviations, Bonferroni and Lorenz curves, and the moments of order statistics. We propose a location-scale regression model based on the log-transmuted Weibull distribution for modeling lifetime data. Applications to two real datasets are given to illustrate the flexibility and potentiality of the transmuted Weibull family of lifetime distributions.     

A family of minimum quantile distance estimators for the three-parameter Weibull distribution

A family of minimum quantile distance estimators, based on a subset of the sample quantiles, is proposed for the parameters of the three-parameter Weibull distribution. The estimation procedure is applicable to either complete or censored samples and, through use of the associated distance measure, provides a goodness-of-fit test for the Weibull model. The proposed estimators are both consistent and asymptotically normal and, in a particular instance, are optimal over the class of all estimators based on the same quantile subset. The problem of optimal quantile selection is also considered.     

The Fisher information matrix for a three-parameter exponentiated Weibull distribution under type II censoring

This paper considers the three-parameter exponentiated Weibull family under type II censoring. It first graphically illustrates the shape property of the hazard function. Then, it proposes a simple algorithm for computing the maximum likelihood estimator and derives the Fisher information matrix. The latter is represented through a single integral in terms of the hazard function; hence it solves the problem of computational difficulty in constructing inferences for the maximum likelihood estimator. Real data analysis is conducted to illustrate the effect of the censoring rate on the maximum likelihood estimation.     

Goodness-of-Fit Tests for the Power-Generalized Weibull Probability Distribution

The power-generalized Weibull probability distribution is very often used in survival analysis mainly because different values of its parameters allow for various shapes of hazard rate such as monotone increasing/decreasing, ∩-shaped, ∪-shaped, or constant. Modified chi-squared tests based on maximum likelihood estimators of parameters that are shown to be -consistent are proposed. Power of these tests against exponentiated Weibull, three-parameter Weibull, and generalized Weibull distributions is studied using Monte Carlo simulations. It is proposed to use the left-tailed rejection region because these tests are biased with respect to the above alternatives if one will use the right-tailed rejection region. It is also shown that power of the McCulloch test investigated can be two or three times higher than that of Nikulin–Rao–Robson test with respect to the alternatives considered if expected cell frequencies are about 5.     

Predicting observables from a general class of distributions

A general class of distributions is proposed to be the underlying population model from which observables are to be predicted using the Bayesian approach. This class of distributions includes, among others, the Weibull, compound Weibull (or three-parameter Burr-type XII), Pareto, beta, Gompertz and compound Gompertz distributions. A proper general prior density function is suggested and the predictive density functions are obtained in the one- and two-sample cases. The informative sample is assumed to be a type II censored sample. Illustrative examples of Weibull (α,β), Burr-type XII (α,β), and Pareto (α,β) distributions are given and compared with the results obtained by previous researchers.     

Parametric Multiplicative Intensities Models Fitted to Bus Motor Failure Data

Classes of two- and three-parameter multiplicative intensities models are presented. These have the advantage that they can be simply fitted by standard software. The two-parameter models include the Weibull and extreme value models, whereas the three-parameter models can have bathtub-shaped forms. The bus motor failure data are reanalysed to show that all five failures of the 191 buses can be adequately described by the same two-parameter multiplicative intensity model based on the squared logarithm of distance travelled in service since previous repair. This is a new member of the multiplicative intensity family, based on the squared logarithm of distance, or of time for survival data.     

Estimation of weibull location

A simulation study was carried out to compare the performances of two different simple estimators of the location parameter for a three-parameter Weibull distribution Both Estimators have been suggested by recent paper in the literature. Bras and mean square error are examined for many different sample-size and shape-parameter-value combinations. Strong evidence of the domination of one estimator over the other is found.     

Three Kolmogorov-Smirnov-type one-sample tests with improved power properties

Some simple point estimators are proposed for the three-parameter Weibull distribution. Both complete and type II censored sampling are considered. The biases and variances of these estimators are studied by Monte Carlo simulation.

Percentage points for the estimator of the shape parameter are also obtained by Monte Carlo simulation, which enables interval estimation and tests of hypotheses to be carried out for the shape parameter.     

M31. Least squares solutions over interval restrictions

In this article, the three-parameter I.G. distribution is standardized with zero mean and unit variance. The third standard moment α₃ is employed as the shape parameter. Tables of the cumulative probability function are given as a function of the standardized variate z, and of the shape parameter, α₃. Various comparisons are made with the lognormal, Weibull, and gamma distributions.