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.     

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.     

Discriminating Between the Log-Normal and Log-Logistic Distributions

Log-normal and log-logistic distributions are often used to analyze lifetime data. For certain ranges of the parameters, the shape of the probability density functions or the hazard functions can be very similar in nature. It might be very difficult to discriminate between the two distribution functions. In this article, we consider the discrimination procedure between the two distribution functions. We use the ratio of maximized likelihood for discrimination purposes. The asymptotic properties of the proposed criterion are investigated. It is observed that the asymptotic distributions are independent of the unknown parameters. The asymptotic distributions are used to determine the minimum sample size needed to discriminate between these two distribution functions for a user specified probability of correct selection. We perform some simulation experiments to see how the asymptotic results work for small sizes. For illustrative purpose, two data sets are analyzed.     

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.     

On a goodness-of-fit test for normality with unknown parameters and type-II censored data

We propose a new goodness-of-fit test for normal and lognormal distributions with unknown parameters and type-II censored data. This test is a generalization of Michael's test for censored samples, which is based on the empirical distribution and a variance stabilizing transformation. We estimate the parameters of the model by using maximum likelihood and Gupta's methods. The quantiles of the distribution of the test statistic under the null hypothesis are obtained through Monte Carlo simulations. The power of the proposed test is estimated and compared to that of the Kolmogorov–Smirnov test also using simulations. The new test is more powerful than the Kolmogorov–Smirnov test in most of the studied cases. Acceptance regions for the PP, QQ and Michael's stabilized probability plots are derived, making it possible to visualize which data contribute to the decision of rejecting the null hypothesis. Finally, an illustrative example is presented.     

Discriminating between distributions using feed-forward neural networks

A relevant problem in many applicatory contexts is to test whether some given observations follow one of two possible probability distributions. The vast literature produced over the years on this topic does not identify a tool which can be easily adopted to any situation but only finds solutions to specific comparisons. Recently, an easy to implement procedure for discrimination between two distributions based on feed-forward neural networks has been proposed giving interesting results. In this work this procedure is further investigated in terms of power, neural network architecture and expected statistical properties of the test statistic for small, moderate and large sample sizes, in a wide range of symmetric and skewed alternatives.     

The estimation of treatment effect for the censored bivariate data under the univariate censoring

This paper establishes a nonparametric estimator for the treatment effect on censored bivariate data under unvariate censoring. This proposed estimator is based on the one from Lin and Ying(1993)'s nonparametric bivariate survival function estimator, which is itself a generalized version of Park and Park(1995)' quantile estimator. A Bahadur type representation of quantile functions were obtained from the marginal survival distribution estimator of Lin and Ying' model. The asymptotic property of this estimator is shown below and the simulation studies are also given     

The Weibull Marshall–Olkin family: Regression model and application to censored data

We introduce a new class of distributions called the Weibull Marshall–Olkin-G family. We obtain some of its mathematical properties. The special models of this family provide bathtub-shaped, decreasing-increasing, increasing-decreasing-increasing, decreasing-increasing-decreasing, monotone, unimodal and bimodal hazard functions. The maximum likelihood method is adopted for estimating the model parameters. We assess the performance of the maximum likelihood estimators by means of two simulation studies. We also propose a new family of linear regression models for censored and uncensored data. The flexibility and importance of the proposed models are illustrated by means of three real data sets.     

Asymptotic distributions of some test statistics for a linear functional relationship

We consider the testing problems of the structural parameters for the multivariate linear functional relationship model. We treat the likelihood ratio test statistics and the test statistics based on the asymptotic distributions of the maximum likelihood estimators. We derive their asymptotic distributions under each null hypothesis respectively. A simulation study is made to evaluate how we can trust our asymptotic results when the sample size is rather small.     

New goodness of fit tests for the Cauchy distribution

Some goodness-of-fit procedures for the Cauchy distribution are presented. The power comparisons indicate that the new tests possess good performances among the competitors, especially against symmetric alternatives. A financial data set is analyzed for illustration.     

Exact inference for a simple step-stress model with Type-II hybrid censored data from the exponential distribution

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors such as temperature, voltage and load on the lifetimes of experimental units. Step-stress test, which is a special class of accelerated life-tests, allows the experimenter to increase the stress levels at fixed times during the experiment in order to obtain information on the parameters of the life distributions more quickly than under normal operating conditions. In this paper, we consider a new step-stress model in which the life-testing experiment gets terminated either at a pre-fixed time (say, _Tm+1$T_{m+1}$) or at a random time ensuring at least a specified number of failures (say, r out of n). Under this model in which the data obtained are Type-II hybrid censored, we consider the case of exponential distribution for the underlying lifetimes. We then derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs and the parametric bootstrap methods, and assess their performance through a Monte Carlo simulation study. Finally, we present two examples to illustrate all the methods of inference discussed here.     

Bayesian inference: Weibull Poisson model for censored data using the expectation–maximization algorithm and its application to bladder cancer data

This article focuses on the parameter estimation of experimental items/units from Weibull Poisson Model under progressive type-II censoring with binomial removals (PT-II CBRs). The expectation–maximization algorithm has been used for maximum likelihood estimators (MLEs). The MLEs and Bayes estimators have been obtained under symmetric and asymmetric loss functions. Performance of competitive estimators have been studied through their simulated risks. One sample Bayes prediction and expected experiment time have also been studied. Furthermore, through real bladder cancer data set, suitability of considered model and proposed methodology have been illustrated.     

Bootstrap likelihood ratio test for Weibull mixture models fitted to grouped data

Abstract

Weibull mixture models are widely used in a variety of fields for modeling phenomena caused by heterogeneous sources. We focus on circumstances in which original observations are not available, and instead the data comes in the form of a grouping of the original observations. We illustrate EM algorithm for fitting Weibull mixture models for grouped data and propose a bootstrap likelihood ratio test (LRT) for determining the number of subpopulations in a mixture model. The effectiveness of the LRT methods are investigated via simulation. We illustrate the utility of these methods by applying them to two grouped data applications.     

On A method for selecting a distributional model

A method for selecting a distributional model for a random variable, given a random sample of observations of it, is studied for various cases. The problems considered include those of choosing between the Weibull and lognormal distributions, between the lognormal and gamma distributions, and between the gamma and Weibull distributions, as well as choosing one of the three. Simulation studies were performed to estimate probabilities of correct selection for the method when it is applied to these problems     

Goodness-of-Fit Test and Parameter Estimation for a Proportional Odds Model of Random Censorship

Censored data arise naturally in a number of fields, particularly in problems of reliability and survival analysis. There are several types of censoring, in this article, we will confine ourselves to the right randomly censoring type. Recently, Ahmadi et al. (2010 Ahmadi , J. , Doostparast , M. , Parsian , A. ( 2010 ). Bayes estimation based on random censored data for some life time models under symmetric and asymmetric loss functions . Communcations in Statistics-Theory and Methods , 39 : 3058 – 3071 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered the problem of estimating unknown parameters in a general framework based on the right randomly censored data. They assumed that the survival function of the censoring time is free of the unknown parameter. This assumption is sometimes inappropriate. In such cases, a proportional odds (PO) model may be more appropriate (Lam and Leung, 2001 Lam , K. F. , Leung , T. L. ( 2001 ). Marginal likelihood estimation for proportional odds models with right censored data . Lifetime Data Analysis 7 : 39 – 54 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Under this model, in this article, point and interval estimations for the unknown parameters are obtained. Since it is important to check the adequacy of models upon which inferences are based (Lawless, 2003 Lawless , J. F. (2003). Statistical Models and Methods for Lifetime Data. , 2nd ed. New York : John Wiley & Sons. [Google Scholar], p. 465), two new goodness-of-fit tests for PO model based on right randomly censored data are proposed. The proposed procedures are applied to two real data sets due to Smith (2002 Smith , P. J. ( 2002 ). Analysis of Failure and Survival Data . London : Chapman & Hall, CRC . [Google Scholar]). A Monte Carlo simulation study is conducted to carry out the behavior of the estimators obtained.     

Goodness-of-fit tests for lifetime distributions based on Type II censored data

In this article, the general test statistic introduced by Alizadeh Noughabi and Balakrishnan [Goodness of fit using a new estimate of Kullback-Leibler information based on Type II censored data. IEEE Trans Reliab. 2015;64:627–635.] is applied for testing goodness of fit of lifetime distributions based on Type II censored data. The test statistic is constructed based on an estimate of Kullback–Leibler (KL) information. We investigate the properties of the proposed test statistic such as the test statistic is nonnegative, just like KL information. We apply this test statistic to following distributions: Exponential, Weibull, Log-normal and Pareto. The critical values and Type I error of the proposed tests are obtained. It is shown that the proposed tests have an excellent Type I error and hence can be used confidently in practice. Then, by Monte Carlo simulations, the power values of the proposed tests are computed against several alternatives and compared with those of the existing tests. Finally, some real-world reliability data are used for illustrative purpose.     

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1–39.], and (ii) an approximation to the one proposed by Barndorff–Nielsen [Barndorff–Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33–53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655–661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff–Nielsen's adjustment.     

A density-based empirical likelihood ratio goodness-of-fit test for the Rayleigh distribution and power comparison

The Rayleigh distribution has been used to model right skewed data. Rayleigh [On the resultant of a large number of vibrations of the some pitch and of arbitrary phase. Philos Mag. 1880;10:73–78] derived it from the amplitude of sound resulting from many important sources. In this paper, a new goodness-of-fit test for the Rayleigh distribution is proposed. This test is based on the empirical likelihood ratio methodology proposed by Vexler and Gurevich [Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy. Comput Stat Data Anal. 2010;54:531–545]. Consistency of the proposed test is derived. It is shown that the distribution of the proposed test does not depend on scale parameter. Critical values of the test statistic are computed, through a simulation study. A Monte Carlo study for the power of the proposed test is carried out under various alternatives. The performance of the test is compared with some well-known competing tests. Finally, an illustrative example is presented and analysed.