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.     

A class of regression models for parallel and series systems with a random number of components

We extend the Weibull power series (WPS) class of distributions to the new class of extended Weibull power series (EWPS) class of distributions. The EWPS distributions are related to series and parallel systems with a random number of components, whereas the WPS distributions [Morais AL, Barreto-Souza W. A compound class of Weibull and power series distributions. Computational Statistics and Data Analysis. 2011;55:1410–1425] are related to series systems only. Unlike the WPS distributions, for which the Weibull is a limiting special case, the Weibull law is a particular case of the EWPS distributions. We prove that the distributions in this class are identifiable under a simple assumption. We also prove stochastic and hazard rate order results and highlight that the shapes of the EWPS distributions are markedly more flexible than the shapes of the WPS distributions. We define a regression model for the EWPS response random variable to model a scale parameter and its quantiles. We present the maximum likelihood estimator and prove its consistency and asymptotic normal distribution. Although series and parallel systems motivated the construction of this class, the EWPS distributions are suitable for modelling a wide range of positive data sets. To illustrate potential uses of this model, we apply it to a real data set on the tensile strength of coconut fibres and present a simple device for diagnostic purposes.     

Maximum log-likelihood ratio test for a change in three parameter Weibull distribution

Asymptotic behavior of a log-likelihood ratio statistic for testing a change in a three parameter Weibull distribution is studied. It is shown that if a shape parameter α>2$\alpha >2$ the law of iterated logarithm for maximum-likelihood estimators is still valid and the log-likelihood testing statistic is asymptotically distributed (after an appropriate normalization) according to a Gumbel distribution.     

A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

Asymptotically Efficient Estimation of a Bivariate Gaussian–Weibull Distribution and an Introduction to the Associated Pseudo-truncated Weibull

Two important wood properties are stiffness (modulus of elasticity or MOE) and bending strength (modulus of rupture or MOR). In the past, MOE has often been modeled as a Gaussian and MOR as a lognormal or a two or three parameter Weibull. It is well known that MOE and MOR are positively correlated. To model the simultaneous behavior of MOE and MOR for the purposes of wood system reliability calculations, we introduce a bivariate Gaussian–Weibull distribution and the associated pseudo-truncated Weibull. We use asymptotically efficient likelihood methods to obtain an estimator of the parameter vector of the bivariate Gaussian–Weibull, and then obtain the asymptotic distribution of this estimator.     

Distance measures in a one-parameter class of density functions

In a class of density functions with parameter λ > 0 , which includes Weibull and Gamma distributions, three distance measures are examined. Several properties of these measures and relationships between them are proved. Furthermore the exponential distribution is characterized in a one- parameter class of Weibull distributions.     

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.     

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 comparison of the parameter estimation methods for bimodal mixture Weibull distribution with complete data

Bimodal mixture Weibull distribution being a special case of mixture Weibull distribution has been used recently as a suitable model for heterogeneous data sets in many practical applications. The bimodal mixture Weibull term represents a mixture of two Weibull distributions. Although many estimation methods have been proposed for the bimodal mixture Weibull distribution, there is not a comprehensive comparison. This paper presents a detailed comparison of five kinds of numerical methods, such as maximum likelihood estimation, least-squares method, method of moments, method of logarithmic moments and percentile method (PM) in terms of several criteria by simulation study. Also parameter estimation methods are applied to real data.     

A Parametric Model for the Interval Censored Survival Times of Acacia Mangium Plantation in a Spacing Trial

Survival times for the Acacia mangium plantation in the Segaliud Lokan Project, Sabah, East Malaysia were analysed based on 20 permanent sample plots (PSPs) established in 1988 as a spacing experiment. The PSPs were established following a complete randomized block design with five levels of spacing randomly assigned to units within four blocks at different sites. The survival times of trees in years are of interest. Since the inventories were only conducted annually, the actual survival time for each tree was not observed. Hence, the data set comprises censored survival times. Initial analysis of the survival of the Acacia mangium plantation suggested there is block by spacing interaction; a Weibull model gives a reasonable fit to the replicate survival times within each PSP; but a standard Weibull regression model is inappropriate because the shape parameter differs between PSPs. In this paper we investigate the form of the non-constant Weibull shape parameter. Parsimonious models for the Weibull survival times have been derived using maximum likelihood methods. The factor selection for the parameters is based on a backward elimination procedure. The models are compared using likelihood ratio statistics. The results suggest that both Weibull parameters depend on spacing and block.     

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.     

A wavelet Whittle estimator of generalized long-memory stochastic volatility

We consider a k-GARMA generalization of the long-memory stochastic volatility model, discuss the properties of the model and propose a wavelet-based Whittle estimator for its parameters. Its consistency is shown. Monte Carlo experiments show that the small sample properties are essentially indistinguishable from those of the Whittle estimator, but are favorable with respect to a wavelet-based approximate maximum likelihood estimator. An application is given for the Microsoft Corporation stock, modeling the intraday seasonal patterns of its realized volatility.     

Business Cycle Duration Dependence Reconsidered

Sichel estimated a Weibull hazard model using the National Bureau of Economic Research business cycle chronology and found evidence of duration dependence only for prewar expansions and postwar contractions. The article updates the postwar sample through the end of the most recent expansion and uses a generalized Weibull model that provides much greater flexibility at the expense of one additional parameter. This model finds evidence of duration dependence for all samples and is statistically superior to the conventional Weibull model for all samples except postwar contractions.     

Bayesian estimation and prediction for the inverse weibull distribution under general progressive censoring

Abstract

This paper deals with Bayesian estimation and prediction for the inverse Weibull distribution with shape parameter α and scale parameter λ under general progressive censoring. We prove that the posterior conditional density functions of α and λ are both log-concave based on the assumption that λ has a gamma prior distribution and α follows a prior distribution with log-concave density. Then, we present the Gibbs sampling strategy to estimate under squared-error loss any function of the unknown parameter vector (α, λ) and find credible intervals, as well as to obtain prediction intervals for future order statistics. Monte Carlo simulations are given to compare the performance of Bayesian estimators derived via Gibbs sampling with the corresponding maximum likelihood estimators, and a real data analysis is discussed in order to illustrate the proposed procedure. Finally, we extend the developed methodology to other two-parameter distributions, including the Weibull, Burr type XII, and flexible Weibull distributions, and also to general progressive hybrid censoring.     

Bayesian Accelerated Life Testing under Competing Weibull Causes of Failure

Consider a J-component series system which is put on Accelerated Life Test (ALT) involving K stress variables. First, a general formulation of ALT is provided for log-location-scale family of distributions. A general stress translation function of location parameter of the component log-lifetime distribution is proposed which can accommodate standard ones like Arrhenius, power-rule, log-linear model, etc., as special cases. Later, the component lives are assumed to be independent Weibull random variables with a common shape parameter. A full Bayesian methodology is then developed by letting only the scale parameters of the Weibull component lives depend on the stress variables through the general stress translation function. Priors on all the parameters, namely the stress coefficients and the Weibull shape parameter, are assumed to be log-concave and independent of each other. This assumption is to facilitate Gibbs sampling from the joint posterior. The samples thus generated from the joint posterior is then used to obtain the Bayesian point and interval estimates of the system reliability at usage condition.     

Approximate prediction limit for weibull failure based on preliminary test estimator

In this paper we have considered type II censored sample from a two parameter Weibull distribution with the known scale parameter. Using the preliminary test estimator of the unknown shape parameter (3 proposed by Pandey (1983), the paper derives a method of finding the approximate prediction limit for the minimum or, more generally,the j^th smallest of a set of future observations from the Weibull or even extreme-value distribution     

Improved inference for the shape-scale family of distributions under type-II censoring

The presence of a nuisance parameter may often perturb the quality of the likelihood-based inference for a parameter of interest under small to moderate sample sizes. The article proposes a maximal scale invariant transformation for likelihood-based inference for the shape in a shape-scale family to circumvent the effect of the nuisance scale parameter. The transformation can be used under complete or type-II censored samples. Simulation-based performance evaluation of the proposed estimator for the popular Weibull, Gamma and Generalized exponential distribution exhibits markedly improved performance in all types of likelihood-based inference for the shape under complete and type-II censored samples. The simulation study leads to a linear relation between the bias of the classical maximum likelihood estimator (MLE) and the transformation-based MLE for the popular Weibull and Gamma distributions. The linearity is exploited to suggest an almost unbiased estimator of the shape parameter for these distributions. Allied estimation of scale is also discussed.     

On extreme order statistics from heterogeneous Weibull variables

In this paper, we focus on stochastic comparisons of extreme order statistics from heterogeneous independent/interdependent Weibull samples. Specifically, we study extreme order statistics from Weibull distributions with (i) common shape parameter but different scale parameters, and (ii) common scale parameter but different shape parameters. Several new comparison results in terms of the likelihood ratio order, reversed hazard rate order and usual stochastic order are studied in those scenarios. The results established here strengthen and generalize some of the results known in the literature including Khaledi and Kochar [Weibull distribution: some stochastic comparisons. J Statist Plann Inference. 2006;136:3121–3129], Fang and Zhang [Stochastic comparisons of series systems with heterogeneous Weibull components. Statist Probab Lett. 2013;83:1649–1653], Torrado [Comparisons of smallest order statistics from Weibull distributions with different scale and shape parameters. J Korean Statist Soc. 2015;44:68–76] and Torrado and Kochar [Stochastic order relations among parallel systems from Weibull distributions. J Appl Probab. 2015;52:102–116]. Some numerical examples are also provided for illustration.     

Markov chain Monte Carlo methods for parameter estimation of the modified Weibull distribution

In this paper, the Markov chain Monte Carlo (MCMC) method is used to estimate the parameters of a modified Weibull distribution based on a complete sample. While maximum-likelihood estimation (MLE) is the most used method for parameter estimation, MCMC has recently emerged as a good alternative. When applied to parameter estimation, MCMC methods have been shown to be easy to implement computationally, the estimates always exist and are statistically consistent, and their probability intervals are convenient to construct. Details of applying MCMC to parameter estimation for the modified Weibull model are elaborated and a numerical example is presented to illustrate the methods of inference discussed in this paper. To compare MCMC with MLE, a simulation study is provided, and the differences between the estimates obtained by the two algorithms are examined.     

Tests for homogeneity of extreme value scale parameters

It is often assumed in situations in which life data from Weibull or extreme-value distributions are involved that data in different samples come from extreme-value distributions with the same scale parameter (equivalently, Weibull distributions with the same shape parameter). This paper proposes a number of tests for homogeneity for extreme-value scale parameters, based on a number of commonly used estimators for these scale parameters. Previous theoretical work and some simulation results provided here indicate that the null distributions of the test statistics proposed are well approximated by the x2 distribution under a wide range of conditions