L-moments of residual life

The mean, variance and coefficient of variation of residual life which are based on the usual moments of the residual life distribution are extensively used in reliability analysis. It has been established in various theoretical and empirical studies that the L-moments have some advantages over the usual moments in many situations. Accordingly in the present paper we study the properties of L-moments of residual life in the context of modeling lifetime data, characterizing life distributions and other applications. The role of certain quantile functions and quantile-based concepts in reliability analysis are also investigated.     

The Govindarajulu Distribution: Some Properties and Applications

In this article, we present various distributional properties and application to reliability analysis of the Govindarajulu distribution. A quantile-based analysis is performed as the distribution function is not analytically tractable. The properties of the distribution like percentiles, L-moments, L-skewness, and kurtosis and order statistics are presented. Various reliability characteristics are derived along with some characterization theorems by relationship between reliability measures. We also make a comparative study with other competing models with reference to real data.     

A skew logistic distribution as an alternative to the model of van Staden and King

The logistic distribution is a simple distribution possessing many useful properties and has been used extensively for analyzing growth. Recently, van Staden and King proposed a quantile-based skew logistic distribution. In this paper, we introduce an alternative skew logistic distribution. We then establish recurrence relations for the computation of the single and product moments of order statistics from the standard skew logistic distribution by using the moments of order statistics from the standard half logistic distribution. These enable an efficient computation of means, variances and covariances of order statistics from the skew logistic distibution for all sample sizes. The results become useful in determining the best linear unbiased estimators of the location and scale paramters of the skew logistic distribution. Finally, we provide an example to illustrate the usefulness of the developed model and then compare its fit with that provided by the model of van Staden and King.     

Quantile-based cumulative Kullback–Leibler divergence

The paper introduces a quantile-based cumulative Kullback–Leibler divergence and study its various properties. Unlike the distribution function approach, the quantile-based measure possesses some unique properties. The quantile functions used in many applied works do not have any tractable distribution functions where the proposed measure is a useful tool to compute the distance between two random variables. Some useful bounds are obtained for quantile-based residual cumulative Kullback–Leibler divergence and quantile-based reliability measures. Characterization results based on the functional forms of quantile-based residual Kullback–Leibler divergence are obtained for some well-known life distributions, namely exponential, Pareto II and beta.     

Theoretical L-moments and TL-moments using combinatorial identities and finite operators

Moments have been traditionally used to characterize a probability distribution. Recently, linear moments (L-moments) and trimmed L-moments (TL-moments) are appealing alternatives to the conventional moments. This paper focuses on the computation of theoretical L-moments and TL-moments and emphasizes the use of combinatorial identities. We are able to derive new closed-form formulas of L-moments and TL-moments for continuous probability distributions. Finally, closed-form formulas for the L-moments for the exponential distribution and the uniform distribution are also obtained.     

Bayesian composite quantile regression

One advantage of quantile regression, relative to the ordinary least-square (OLS) regression, is that the quantile regression estimates are more robust against outliers and non-normal errors in the response measurements. However, the relative efficiency of the quantile regression estimator with respect to the OLS estimator can be arbitrarily small. To overcome this problem, composite quantile regression methods have been proposed in the literature which are resistant to heavy-tailed errors or outliers in the response and at the same time are more efficient than the traditional single quantile-based quantile regression method. This paper studies the composite quantile regression from a Bayesian perspective. The advantage of the Bayesian hierarchical framework is that the weight of each component in the composite model can be treated as open parameter and automatically estimated through Markov chain Monte Carlo sampling procedure. Moreover, the lasso regularization can be naturally incorporated into the model to perform variable selection. The performance of the proposed method over the single quantile-based method was demonstrated via extensive simulations and real data analysis.     

Comparisons of ten estimation methods for the parameters of Marshall–Olkin extended exponential distribution

The aim of this article is to compare via Monte Carlo simulations the finite sample properties of the parameter estimates of the Marshall–Olkin extended exponential distribution obtained by ten estimation methods: maximum likelihood, modified moments, L-moments, maximum product of spacings, ordinary least-squares, weighted least-squares, percentile, Crámer–von-Mises, Anderson–Darling, and Right-tail Anderson–Darling. The bias, root mean-squared error, absolute and maximum absolute difference between the true and estimated distribution functions are used as criterion of comparison. The simulation study reveals that the L-moments and maximum products of spacings methods are highly competitive with the maximum likelihood method in small as well as in large-sized samples.     

Maximum likelihood computation for fitting semiparametric mixture models

Three general algorithms that use different strategies are proposed for computing the maximum likelihood estimate of a semiparametric mixture model. They seek to maximize the likelihood function by, respectively, alternating the parameters, profiling the likelihood and modifying the support set. All three algorithms make a direct use of the recently proposed fast and stable constrained Newton method for computing the nonparametric maximum likelihood of a mixing distribution and employ additionally an optimization algorithm for unconstrained problems. The performance of the algorithms is numerically investigated and compared for solving the Neyman-Scott problem, overcoming overdispersion in logistic regression models and fitting two-level mixed effects logistic regression models. Satisfactory results have been obtained.     

Gini index based goodness-of-fit test for the logistic distribution

To model growth curves in survival analysis and biological studies the logistic distribution has been widely used. In this article, we propose a goodness-of-fit test for the logistic distribution based on an estimate of the Gini index. The exact distribution of the proposed test statistic and also its asymptotic distribution are presented. In order to compute the proposed test statistic, parameters of the logistic distribution are estimated by approximate maximum likelihood estimators (AMLEs), which are simple explicit estimators. Through Monte Carlo simulations, power comparisons of the proposed test with some known competing tests are carried. Finally, an illustrative example is presented and analyzed.     

A New Quantile Function with Applications to Reliability Analysis

In this article, we propose a new class of distributions defined by a quantile function, which nests several distributions as its members. The quantile function proposed here is the sum of the quantile functions of the generalized Pareto and Weibull distributions. Various distributional properties and reliability characteristics of the class are discussed. The estimation of the parameters of the model using L-moments is studied. Finally, we apply the model to a real life dataset.     

Chi-Square quantile-based multivariate variance monitoring for individual observations

Under the normality assumption, some statistics for monitoring a multivariate process variance for individual observations can be used to detect a variance shift, but the distribution of their in-control run length has a high variance as well as the median that is extremely smaller than the mean, which leads to many false alarms in the in-control process. In this paper, we propose a chi-square quantile-based monitoring statistic which is free of the problems. The numerical experiments show that the proposed monitoring statistics outperform the existing monitoring statistics in terms of the detection of a shift for the variance.     

The complementary beta distribution

The complementary beta distribution is proposed as a new distribution on the unit interval. It results from reversing the roles of the distribution and quantile functions of the beta distribution. It has some attractive properties that are complementary to those of the beta distribution. In particular, the complementary beta distribution is much more amenable than the beta distribution to exact computations involving expectations of order statistics, including L-moments. At least for a wide range of parameter values, complementary beta and beta distributions with parameters that are reciprocals of the other's parameters are good approximations to one another. We also note the position of the complementary beta distribution in a wider family of distributions defined through the same simple form for their quantile density functions.     

Simulation-based Inference in a Zero-inflated Bernoulli Regression Model

The logistic regression model has become a standard tool to investigate the relationship between a binary outcome and a set of potential predictors. When analyzing binary data, it often arises that the observed proportion of zeros is greater than expected under the postulated logistic model. Zero-inflated binomial (ZIB) models have been developed to fit binary data that contain too many zeros. Maximum likelihood estimators in these models have been proposed and their asymptotic properties established. Several aspects of ZIB models still deserve attention however, such as the estimation of odds-ratios and event probabilities. In this article, we propose estimators of these quantities and we investigate their properties both theoretically and via simulations. Based on these results, we provide recommendations about the range of conditions (minimum sample size, maximum proportion of zeros in excess) under which a reliable statistical inference on the odds-ratios and event probabilities can be obtained in a ZIB regression model. A real-data example illustrates the proposed estimators.     

A maximin criterion for the logistic random intercept model with covariates

A maximin criterion is used to find optimal designs for the logistic random intercept model with dichotomous independent variables. The dichotomous independent variables can be subdivided into variables for which the distribution is specified prior to data sampling, called variates, and into variables for which the distribution is not specified prior to data sampling, but is obtained from data sampling, called covariates. The proposed maximin criterion maximizes the smallest possible relative efficiency not only with respect to all possible values of the model parameters, but also with respect to the joint distribution of the covariates. We have shown that, under certain conditions, the maximin design is balanced with respect to the joint distribution of the variates. The proposed method will be used to plan a (stratified) clinical trial where variates and covariates are involved.     

On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F ^α(x)(1 ? F(x))^β, α, β ? , is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and—for the first time—transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom and, newly, the distribution corresponding to α = β = 5/2. The logistic distribution is central to some of the developments of the article.     

Estimation of the generalized logistic distribution parameters: Comparative study

The generalized logistic distribution can be considered as a proportional reversed hazard family with the baseline distribution as the logistic distribution. The generalized logistic distribution has been used to model the data with a unimodal density. In this comparison paper, the authors considered the maximum likelihood estimation of the different parameters of a generalized logistic distribution as well as other five estimation procedures. In this paper, we compare the performances of these procedures through an extensive numerical simulation.     

Quantile based tests for exponentiality against DMRQ and NBUE alternatives

Quantile-based reliability analysis has received much attention recently. We propose new quantile-based tests for exponentiality against decreasing mean residual quantile function (DMRQ) and new better than used in expectation (NBUE) classes of alternatives. The exact null distribution of the test statistic is derived when the alternative class is DMRQ. The asymptotic properties of both the test statistics are studied. The performance of the proposed tests with other existing tests in the literature is evaluated through simulation study. Finally, we illustrate our test procedure using real data sets.     

An extension of the Freund's bivariate distribution to model cascading failures

The notion of cascading failures is a common phenomenon we observe around us. Here the initial failure alters the structure function of the system, which leads to subsequent failures within a short period of time referred to as threshold time. The concept of cascading failures within the framework of reliability theory and the Freund bivariate exponential distribution to model cascading failures has been studied by few authors. The Freund bivariate exponential distribution allows modelling a parallel redundant system with two components. In this system, the lifetimes of the two components behave as if they are independent, until one of the components fail, after which the remaining component suffers an increased/decreased stress. In this article, we further generalize this model to accommodate cascading failures. Various properties of the model are investigated and statistical inference procedures are developed using L-moments and method of moments. A practical application of this model is illustrated using data from www.espncricinfo.com. Also well analysed Diabetic Retinopathy Study (DRS) data is further analysed using this model and our findings are presented.     

Logistic regression in meta-analysis using aggregate data

We derived two methods to estimate the logistic regression coefficients in a meta-analysis when only the 'aggregate' data (mean values) from each study are available. The estimators we proposed are the discriminant function estimator and the reverse Taylor series approximation. These two methods of estimation gave similar estimators using an example of individual data. However, when aggregate data were used, the discriminant function estimators were quite different from the other two estimators. A simulation study was then performed to evaluate the performance of these two estimators as well as the estimator obtained from the model that simply uses the aggregate data in a logistic regression model. The simulation study showed that all three estimators are biased. The bias increases as the variance of the covariate increases. The distribution type of the covariates also affects the bias. In general, the estimator from the logistic regression using the aggregate data has less bias and better coverage probabilities than the other two estimators. We concluded that analysts should be cautious in using aggregate data to estimate the parameters of the logistic regression model for the underlying individual data.