Planning step-stress test plans under Type-I censoring for the log-location-scale case

In this paper, we consider a k-level step-stress accelerated life-testing (ALT) experiment with unequal duration steps τ=(τ₁, …, τ _k ). Censoring is allowed only at the change-stress point in the final stage. A general log-location-scale lifetime distribution with mean life which is a linear function of stress, along with a cumulative exposure model, is considered as the working model. Under this model, the determination of the optimal choice of τ for both Weibull and lognormal distributions are addressed using the variance–optimality criterion. Numerical results show that for a general log-location-scale distributions, the optimal k-step-stress ALT model with unequal duration steps reduces just to a 2-level step-stress ALT model.     

Two new defective distributions based on the Marshall–Olkin extension

The presence of immune elements (generating a fraction of cure) in survival data is common. These cases are usually modeled by the standard mixture model. Here, we use an alternative approach based on defective distributions. Defective distributions are characterized by having density functions that integrate to values less than \(1\), when the domain of their parameters is different from the usual one. We use the Marshall–Olkin class of distributions to generalize two existing defective distributions, therefore generating two new defective distributions. We illustrate the distributions using three real data sets.     

Parameter estimation for mixtures of skew Laplace normal distributions and application in mixture regression modeling

In this article, we propose mixtures of skew Laplace normal (SLN) distributions to model both skewness and heavy-tailedness in the neous data set as an alternative to mixtures of skew Student-t-normal (STN) distributions. We give the expectation–maximization (EM) algorithm to obtain the maximum likelihood (ML) estimators for the parameters of interest. We also analyze the mixture regression model based on the SLN distribution and provide the ML estimators of the parameters using the EM algorithm. The performance of the proposed mixture model is illustrated by a simulation study and two real data examples.     

The minimum coverage probability of confidence intervals in regression after a preliminary F test

Consider a linear regression model with regression parameter β=(β₁,…,β_p) and independent normal errors. Suppose the parameter of interest is θ=a^Tβ, where a is specified. Define the s-dimensional parameter vector τ=C^Tβ−t, where C and t are specified. Suppose that we carry out a preliminary F test of the null hypothesis H₀:τ=0 against the alternative hypothesis H₁:τ≠0. It is common statistical practice to then construct a confidence interval for θ with nominal coverage 1−α, using the same data, based on the assumption that the selected model had been given to us a priori (as the true model). We call this the naive 1−α confidence interval for θ. This assumption is false and it may lead to this confidence interval having minimum coverage probability far below 1−α, making it completely inadequate. We provide a new elegant method for computing the minimum coverage probability of this naive confidence interval, that works well irrespective of how large s is. A very important practical application of this method is to the analysis of covariance. In this context, τ can be defined so that H₀ expresses the hypothesis of “parallelism”. Applied statisticians commonly recommend carrying out a preliminary F test of this hypothesis. We illustrate the application of our method with a real-life analysis of covariance data set and a preliminary F test for “parallelism”. We show that the naive 0.95 confidence interval has minimum coverage probability 0.0846, showing that it is completely inadequate.     

Robust mixture modeling using the skew <Emphasis Type="Italic">t</Emphasis> distribution

A finite mixture model using the Student's t distribution has been recognized as a robust extension of normal mixtures. Recently, a mixture of skew normal distributions has been found to be effective in the treatment of heterogeneous data involving asymmetric behaviors across subclasses. In this article, we propose a robust mixture framework based on the skew t distribution to efficiently deal with heavy-tailedness, extra skewness and multimodality in a wide range of settings. Statistical mixture modeling based on normal, Student's t and skew normal distributions can be viewed as special cases of the skew t mixture model. We present analytically simple EM-type algorithms for iteratively computing maximum likelihood estimates. The proposed methodology is illustrated by analyzing a real data example.     

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.     

Forecasting Vector ARMA Processes With Systematically Missing Observations

The following two predictors are compared for time series with systematically missing observations: (a) A time series model is fitted to the full series X_t , and forecasts are based on this model, (b) A time series model is fitted to the series with systematically missing observations Y _τ, and forecasts are based on the resulting model. If the data generation processes are known vector autoregressive moving average (ARMA) processes, the first predictor is at least as efficient as the second one in a mean squared error sense. Conditions are given for the two predictors to be identical. If only the ARMA orders of the generation processes are known and the coefficients are estimated, or if the process orders and coefficients are estimated, the first predictor is again, in general, superior. There are, however, exceptions in which the second predictor, using seemingly less information, may be better. These results are discussed, using both asymptotic theory and small sample simulations. Some economic time series are used as illustrative examples.     

Nonparametric Mixtures Based on Skew-normal Distributions: An Application to Density Estimation

This article addresses the density estimation problem using nonparametric Bayesian approach. It is considered hierarchical mixture models where the uncertainty about the mixing measure is modeled using the Dirichlet process. The main goal is to build more flexible models for density estimation. Extensions of the normal mixture model via Dirichlet process previously introduced in the literature are twofold. First, Dirichlet mixtures of skew-normal distributions are considered, say, in the first stage of the hierarchical model, the normal distribution is replaced by the skew-normal one. We also assume a skew-normal distribution as the center measure in the Dirichlet mixture of normal distributions. Some important results related to Bayesian inference in the location-scale skew-normal family are introduced. In particular, we obtain the stochastic representations for the full conditional distributions of the location and skewness parameters. The algorithm introduced by MacEachern and Müller in 1998 MacEachern, S.N., Müller, P. (1998). Estimating mixture of Dirichlet Process models. J. Computat. Graph. Statist. 7(2):223–238.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar] is used to sample from the posterior distributions. The models are compared considering simulated data sets. Finally, the well-known Old Faithful Geyser data set is analyzed using the proposed models and the Dirichlet mixture of normal distributions. The model based on Dirichlet mixture of skew-normal distributions captured the data bimodality and skewness shown in the empirical distribution.     

A comparative study of the K-means algorithm and the normal mixture model for clustering: Bivariate homoscedastic case

The K-means algorithm and the normal mixture model method are two common clustering methods. The K-means algorithm is a popular heuristic approach which gives reasonable clustering results if the component clusters are ball-shaped. Currently, there are no analytical results for this algorithm if the component distributions deviate from the ball-shape. This paper analytically studies how the K-means algorithm changes its classification rule as the normal component distributions become more elongated under the homoscedastic assumption and compares this rule with that of the Bayes rule from the mixture model method. We show that the classification rules of both methods are linear, but the slopes of the two classification lines change in the opposite direction as the component distributions become more elongated. The classification performance of the K-means algorithm is then compared to that of the mixture model method via simulation. The comparison, which is limited to two clusters, shows that the K-means algorithm provides poor classification performances consistently as the component distributions become more elongated while the mixture model method can potentially, but not necessarily, take advantage of this change and provide a much better classification performance.     

Flexible mixture modelling using the multivariate skew-t-normal distribution

This paper presents a robust probabilistic mixture model based on the multivariate skew-t-normal distribution, a skew extension of the multivariate Student’s t distribution with more powerful abilities in modelling data whose distribution seriously deviates from normality. The proposed model includes mixtures of normal, t and skew-normal distributions as special cases and provides a flexible alternative to recently proposed skew t mixtures. We develop two analytically tractable EM-type algorithms for computing maximum likelihood estimates of model parameters in which the skewness parameters and degrees of freedom are asymptotically uncorrelated. Standard errors for the parameter estimates can be obtained via a general information-based method. We also present a procedure of merging mixture components to automatically identify the number of clusters by fitting piecewise linear regression to the rescaled entropy plot. The effectiveness and performance of the proposed methodology are illustrated by two real-life examples.     

Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.     

On the discrete analogues of continuous distributions

In this paper, a new method is proposed for generating discrete distributions. A special class of the distributions, namely, the T-geometric family contains the discrete analogues of continuous distributions. Some general properties of the T-geometric family of distributions are obtained. A member of the T-geometric family, namely, the exponentiated-exponential–geometric distribution is defined and studied. Various properties of the exponentiated-exponential–geometric distribution such as the unimodality, the moments and the probability generating function are discussed. The method of maximum likelihood estimation is proposed for estimating the model parameters. Three real data sets are used to illustrate the applications of the exponentiated-exponential–geometric distribution.     

Flexible mixture modeling via the multivariate t distribution with?the?Box-Cox transformation: an?alternative to?the?skew-t distribution

Cluster analysis is the automated search for groups of homogeneous observations in a data set. A popular modeling approach for clustering is based on finite normal mixture models, which assume that each cluster is modeled as a multivariate normal distribution. However, the normality assumption that each component is symmetric is often unrealistic. Furthermore, normal mixture models are not robust against outliers; they often require extra components for modeling outliers and/or give a poor representation of the data. To address these issues, we propose a new class of distributions, multivariate t distributions with the Box-Cox transformation, for mixture modeling. This class of distributions generalizes the normal distribution with the more heavy-tailed t distribution, and introduces skewness via the Box-Cox transformation. As a result, this provides a unified framework to simultaneously handle outlier identification and data transformation, two interrelated issues. We describe an Expectation-Maximization algorithm for parameter estimation along with transformation selection. We demonstrate the proposed methodology with three real data sets and simulation studies. Compared with a wealth of approaches including the skew-t mixture model, the proposed t mixture model with the Box-Cox transformation performs favorably in terms of accuracy in the assignment of observations, robustness against model misspecification, and selection of the number of components.     

On the discrete analogues of continuous distributions

In this paper, a new method is proposed for generating discrete distributions. A special class of the distributions, namely, the T-geometric family contains the discrete analogues of continuous distributions. Some general properties of the T-geometric family of distributions are obtained. A member of the T-geometric family, namely, the exponentiated-exponential–geometric distribution is defined and studied. Various properties of the exponentiated-exponential–geometric distribution such as the unimodality, the moments and the probability generating function are discussed. The method of maximum likelihood estimation is proposed for estimating the model parameters. Three real data sets are used to illustrate the applications of the exponentiated-exponential–geometric distribution.     

A mixture model with Poisson and zero-truncated Poisson components to analyze road traffic accidents in Turkey

The analysis of traffic accident data is crucial to address numerous concerns, such as understanding contributing factors in an accident''s chain-of-events, identifying hotspots, and informing policy decisions about road safety management. The majority of statistical models employed for analyzing traffic accident data are logically count regression models (commonly Poisson regression) since a count – like the number of accidents – is used as the response. However, features of the observed data frequently do not make the Poisson distribution a tenable assumption. For example, observed data rarely demonstrate an equal mean and variance and often times possess excess zeros. Sometimes, data may have heterogeneous structure consisting of a mixture of populations, rather than a single population. In such data analyses, mixtures-of-Poisson-regression models can be used. In this study, the number of injuries resulting from casualties of traffic accidents registered by the General Directorate of Security (Turkey, 2005–2014) are modeled using a novel mixture distribution with two components: a Poisson and zero-truncated-Poisson distribution. Such a model differs from existing mixture models in literature where the components are either all Poisson distributions or all zero-truncated Poisson distributions. The proposed model is compared with the Poisson regression model via simulation and in the analysis of the traffic data.     

The n.s. conditions for the ration of two quadratic forms to have an f-distribution and its applications

Suppose that ξ and η be two random vectors and that (ξ^τ, η^τ have an elliptically contoured distribution or a multivariate normal distribution. In this article, we obtain some necessary and sufficient (N.S.) conditions such that the ratio of two quadratic forms, say ξ^τ Aξ and η^τ Bη(for some symmetric nonnegative matrices A and B), has an F-distribution. As applications, we extend the classical F-test to some dependent two group samples. Two cases are considered: elliptically contoured and normal distributions.     

Estimation of extreme quantiles from heavy and light tailed distributions

In Gardes et al. (2011), a new family of distributions is introduced, depending on two parameters τ$\tau$ and θ$\theta$, which encompasses Pareto-type distributions as well as Weibull tail-distributions. Estimators for θ$\theta$ and extreme quantiles are also proposed, but they both depend on the unknown parameter τ$\tau$, making them useless in practical situations. In this paper, we propose an estimator of τ$\tau$ which is independent of θ$\theta$. Plugging our estimator of τ$\tau$ in the two previous ones allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a small simulation study and on a real data set.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

Bayesian inference for the generalized exponential distribution

The two-parameter generalized exponential (GE) distribution was introduced by Gupta and Kundu [Gupta, R.D. and Kundu, D., 1999, Generalized exponential distribution. Australian and New Zealand Journal of Statistics, 41(2), 173–188.]. It was observed that the GE can be used in situations where a skewed distribution for a nonnegative random variable is needed. In this article, the Bayesian estimation and prediction for the GE distribution, using informative priors, have been considered. Importance sampling is used to estimate the parameters, as well as the reliability function, and the Gibbs and Metropolis samplers data sets are used to predict the behavior of further observations from the distribution. Two data sets are used to illustrate the Bayesian procedure.     

The beta Weibull-geometric distribution

A new five-parameter distribution called the beta Weibull-geometric (BWG) distribution is proposed. The new distribution is generated from the logit of a beta random variable and includes the Weibull-geometric distribution of Barreto-Souza et al. [The Weibull-geometric distribution, J. Stat. Comput. Simul. 81 (2011), pp. 645–657], beta Weibull (BW), beta exponential, exponentiated Weibull, and some other lifetime distributions as special cases. A comprehensive mathematical treatment of this distribution is provided. The density function can be expressed as an infinite mixture of BW densities and then we derive some mathematical properties of the new distribution from the corresponding properties of the BW distribution. The density function of the order statistics and also estimation of the stress–strength parameter are obtained using two general expressions. To estimate the model parameters, we use the maximum likelihood method and the asymptotic distribution of the estimators is also discussed. The capacity of the new distribution are examined by various tools, using two real data sets.