The three-parameter two-piece normal family of distributions and its fitting

This note introduces a family of skew and symmetric distributions containing the normal family and indexed by three parameters with clear meanings. Another respect in which this family compares favourably with families like the Pearson family, the Bessel-Gram-Charlier family and the Johnson family is ease of maximum likelihood fitting. Fitting by the method of moments is also considered. Asymptotic distributions of maximum likelihood and moment estimators are worked out. A test of symmetry and normality is suggested.     

Posterior analysis,prediction and reliability in three-parameter weibull distributions

The paper develops Bayesian analysis in the context of samples from three-parameter Weibull distributions and shows how to tackle the problems of prediction and estimation of reliability curves. As Johnson, Kotz and Balakrishnan ( 1994 ) mentioned, the prediction problems for the three-parameter Weibull model seem to be unresolved and is certainly worth looking into (p.671). Posterior analysis organized around Gibbs sampling is shown to perform well. An application to stock returns is used to illustrate the potential of the approach.     

On the comparison of Fisher information of the Weibull and GE distributions

In this paper, we consider the Fisher information matrices of the generalized exponential (GE) and Weibull distributions for complete and Type-I censored observations. Fisher information matrix can be used to compute asymptotic variances of the different estimators. Although both distributions may provide similar data fit but the corresponding Fisher information matrices can be quite different. Moreover, the percentage loss of information due to truncation of the Weibull distribution is much more than the GE distribution. We compute the total information of the Weibull and GE distributions for different parameter ranges. We compare the asymptotic variances of the median estimators and the average asymptotic variances of all the percentile estimators for complete and Type-I censored observations. One data analysis has been preformed for illustrative purposes. When two fitted distributions are very close to each other and very difficult to discriminate otherwise, the Fisher information or the above mentioned asymptotic variances may be used for discrimination purposes.     

On characteristic functions of the wrapped lognormal and the wrapped Weibull distributions

The characteristic function plays a prominent role in determining the pdf of a circular model using the trigonometric moments. The characteristic functions of the wrapped lognormal and the wrapped Weibull distributions cannot be expressed in a closed form. Hence, numerical evaluation of the same is presented along with graphs. Also, certain population characteristics of the wrapped lognormal and the wrapped Weibull distributions are presented.     

Discriminating among Weibull,log-normal,and log-logistic distributions

In this article, we consider the problem of the model selection/discrimination among three different positively skewed lifetime distributions. All these three distributions, namely; the Weibull, log-normal, and log-logistic, have been used quite effectively to analyze positively skewed lifetime data. In this article, we have used three different methods to discriminate among these three distributions. We have used the maximized likelihood method to choose the correct model and computed the asymptotic probability of correct selection. We have further obtained the Fisher information matrices of these three different distributions and compare them for complete and censored observations. These measures can be used to discriminate among these three distributions. We have also proposed to use the Kolmogorov–Smirnov distance to choose the correct model. Extensive simulations have been performed to compare the performances of the three different methods. It is observed that each method performs better than the other two for some distributions and for certain range of parameters. Further, the loss of information due to censoring are compared for these three distributions. The analysis of a real dataset has been performed for illustrative purposes.     

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.     

Discriminating between the generalized Rayleigh and Weibull distributions

Generalized Rayleigh (GR) and Weibull (WE) distributions are used quite effectively for analysing skewed lifetime data. In this paper, we consider the problem of selecting either GR or WE distribution as a more appropriate fitting model for a given data set. We use the ratio of maximized likelihoods (RML) for discriminating between the two distributions. The asymptotic and simulated distributions of the logarithm of the RML are applied to determine the probability of correctly selecting between these two families of distributions. It is examined numerically that the asymptotic results work quite well even for small sample sizes. A real data set involving the annual rainfall recorded at Los Angeles Civic Center during 25 years is analysed to illustrate the procedures developed here.     

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.     

Parameter estimation of three-parameter Weibull distribution based on progressively Type-II censored samples

In this paper, the estimation of parameters for a three-parameter Weibull distribution based on progressively Type-II right censored sample is studied. Different estimation procedures for complete sample are generalized to the case with progressively censored data. These methods include the maximum likelihood estimators (MLEs), corrected MLEs, weighted MLEs, maximum product spacing estimators and least squares estimators. We also proposed the use of a censored estimation method with one-step bias-correction to obtain reliable initial estimates for iterative procedures. These methods are compared via a Monte Carlo simulation study in terms of their biases, root mean squared errors and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate all of the methods of inference developed here.     

Maximum likelihood vs. maximum goodness of fit estimation of the three-parameter Weibull distribution

The use of statistics based on the empirical distribution function is analysed for estimation of the scale, shape, and location parameters of the three-parameter Weibull distribution. The resulting maximum goodness of fit (MGF) estimators are compared with their maximum likelihood counterparts. In addition to the Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling statistics, some related empirical distribution function statistics using different weight functions are considered. The results show that the MGF estimators of the scale and shape parameters are usually more efficient than the maximum likelihood estimators when the shape parameter is smaller than 2, particularly if the sample size is large.     

Discrimination between nested two- and three-parameter air pollutant frequency distributions

The purpose of this paper is to consider methods which discriminate between 2- and 3-parameter nested alternatives for the gamma, Weibull and log-normal distributions, and to investigate their utility in representing frequency distributions of air pollutant measurements. Monte Carlo experiments are conducted to evaluate the likelihood ratio test, Akaike's information criterion, Schwarz's information criterion, the Chi-square test and the Kolmogorov-Smirnov test. The performance of the tests and criteria depends on the types of nested distributions under consideration, the parametric values of the parent distributions, the confidence levels used (if applicable) and the sample sizes. The practical usefulness of the techniques is illustrated by observing the errors of the models in fitting the upper percentiles of the parent distribution. Two sets of air pollution data, namely hourly pollutant observations of B-scattering and nitrogen dioxide, from an urban airshed are used to examine the similarities and differences in fitting 2- and 3-parameter distributions where historical practice suggests there is a preference for the more parsimonious model.     

Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.     

A general class of flexible Weibull distributions

We consider a linear combination of two logarithms of cumulative hazard functions and propose a general class of flexible Weibull distribution functions which includes some well-known modified Weibull distributions (MWDs). We suggest a very flexible Weibull distribution, which belongs to the class, and show that its hazard function is monotone, bathtub-shaped, modified bathtub-shaped, or even upside-down bathtub-shaped. We also discuss the methods of least square estimation and maximum likelihood estimation of the unknown parameters. We take two illustrated examples to compare the suggested distribution with some current MWDs, and show that the suggested distribution shows good performances.     

Extended Weibull type distribution and finite mixture of distributions

An extended form of Weibull distribution is suggested which has two shape parameters (m and δ). Introduction of another shape parameter δ helps to express the extended Weibull distribution not only as an exact form of a mixture of distributions under certain conditions, but also provides extra flexibility to the density function over positive range. The shape of density function of the extended Weibull type distribution for various values of the parameters is shown which may be of some interest to Bayesians. Certain statistical properties such as hazard rate function, mean residual function, rth moment are defined explicitly. The proposed extended Weibull distribution is used to derive an exact form of two, three and k-component mixture of distributions. With the help of a real data set, the usefulness of mixture Weibull type distribution is illustrated by using Markov Chain Monte Carlo (MCMC), Gibbs sampling approach.     

Existence,uniqueness and consistency of estimation of life characteristics of three-parameter Weibull distribution based on Type-II right censored data

In this paper, we propose a new method of estimation for the parameters and quantiles of the three-parameter Weibull distribution based on Type-II right censored data. The method, based on a data transformation, overcomes the problem of unbounded likelihood. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators are also consistent over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method of estimation performs well compared to some prominent methods in terms of bias and root mean squared error in small-sample situations. Finally, two real data sets are used to illustrate the proposed method of estimation.     

Inverse Weibull power series distributions: properties and applications

Inverse Weibull (IW) distribution is one of the widely used probability distributions for nonnegative data modelling, specifically, for describing degradation phenomena of mechanical components. In this paper, by compounding IW and power series distributions we introduce a new lifetime distribution. The compounding procedure follows the same set-up carried out by Adamidis and Loukas [A lifetime distribution with decreasing failure rate. Stat Probab Lett. 1998;39:35–42]. We provide mathematical properties of this new distribution such as moments, estimation by maximum likelihood with censored data, inference for a large sample and the EM algorithm to determine the maximum likelihood estimates of the parameters. Furthermore, we characterize the proposed distributions using a simple relationship between two truncated moments and maximum entropy principle under suitable constraints. Finally, to show the flexibility of this type of distributions, we demonstrate applications of two real data sets.     

Discriminating between the Weibull and log-normal distributions for Type-II censored data

Log-normal and Weibull distributions are the two most popular distributions for analysing lifetime data. In this paper, we consider the problem of discriminating between the two distribution functions. It is assumed that the data are coming either from log-normal or Weibull distributions and that they are Type-II censored. We use the difference of the maximized log-likelihood functions, in discriminating between the two distribution functions. We obtain the asymptotic distribution of the discrimination statistic. It is used to determine the probability of correct selection in this discrimination process. We perform some simulation studies to observe how the asymptotic results work for different sample sizes and for different censoring proportions. It is observed that the asymptotic results work quite well even for small sizes if the censoring proportions are not very low. We further suggest a modified discrimination procedure. Two real data sets are analysed for illustrative purposes.     

The unit extended Weibull families of distributions and its applications

In this paper, two new general families of distributions supported on the unit interval are introduced. The proposed families include several known models as special cases and define at least twenty (each one) new special models. Since the list of well-being indicators may include several double bounded random variables, the applicability for modeling those is the major practical motivation for introducing the distributions on those families. We propose a parametrization of the new families in terms of the median and develop a shiny application to provide interactive density shape illustrations for some special cases. Various properties of the introduced families are studied. Some special models in the new families are discussed. In particular, the complementary unit Weibull distribution is studied in some detail. The method of maximum likelihood for estimating the model parameters is discussed. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Applications to the literacy rate in Brazilian and Colombian municipalities illustrate the usefulness of the two new families for modeling well-being indicators.     

Bayes estimation for the Block and Basu bivariate and multivariate Weibull distributions

Block and Basu bivariate exponential distribution is one of the most popular absolute continuous bivariate distributions. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. Statist Methodol. 2010;7:464–477] introduced Block and Basu bivariate Weibull (BBBW) distribution, which is a generalization of the Block and Basu bivariate exponential distribution, and provided the maximum likelihood estimators using EM algorithm. In this paper, we consider the Bayesian inference of the unknown parameters of the BBBW distribution. The Bayes estimators are obtained with respect to the squared error loss function, and the prior distributions allow for prior dependence among the unknown parameters. Prior independence also can be obtained as a special case. It is observed that the Bayes estimators of the unknown parameters cannot be obtained in explicit forms. We propose to use the importance sampling technique to compute the Bayes estimates and also to construct the associated highest posterior density credible intervals. The analysis of two data sets has been performed for illustrative purposes. The performances of the proposed estimators are quite satisfactory. Finally, we generalize the results for the multivariate case.     

Nonparametric Bayesian survival analysis using mixtures of Weibull distributions

Bayesian nonparametric methods have been applied to survival analysis problems since the emergence of the area of Bayesian nonparametrics. However, the use of the flexible class of Dirichlet process mixture models has been rather limited in this context. This is, arguably, to a large extent, due to the standard way of fitting such models that precludes full posterior inference for many functionals of interest in survival analysis applications. To overcome this difficulty, we provide a computational approach to obtain the posterior distribution of general functionals of a Dirichlet process mixture. We model the survival distribution employing a flexible Dirichlet process mixture, with a Weibull kernel, that yields rich inference for several important functionals. In the process, a method for hazard function estimation emerges. Methods for simulation-based model fitting, in the presence of censoring, and for prior specification are provided. We illustrate the modeling approach with simulated and real data.