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.     

Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variates

We introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. First, we perform a test of symmetry, and second, we estimate the parameters by minimizing the quadratic distance between the real parts of the empirical and theoretical characteristic functions. The quadratic distance estimator obtained is consistent, robust and asymptotically normally distributed. We develop a statistical test for goodness of fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.     

Multivariate log-skewed distributions with normal kernel and their applications

We introduce two classes of multivariate log-skewed distributions with normal kernel: the log canonical fundamental skew-normal (log-CFUSN) and the log unified skew-normal. We also discuss some properties of the log-CFUSN family of distributions. These new classes of log-skewed distributions include the log-normal and multivariate log-skew normal families as particular cases. We discuss some issues related to Bayesian inference in the log-CFUSN family of distributions, mainly we focus on how to model the prior uncertainty about the skewing parameter. Based on the stochastic representation of the log-CFUSN family, we propose a data augmentation strategy for sampling from the posterior distributions. This proposed family is used to analyse the US national monthly precipitation data. We conclude that a high-dimensional skewing function lead to a better model fit.     

The generalized odd log-logistic family of distributions: properties,regression models and applications

We propose a new class of continuous distributions with two extra shape parameters named the generalized odd log-logistic family of distributions. The proposed family contains as special cases the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, two entropy measures and order statistics are obtained. We derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behaviour of the estimators by means of Monte Carlo simulations. We introduce the log-odd log-logistic Weibull regression model with censored data based on the odd log-logistic-Weibull distribution. The importance of the new family is illustrated using three real data sets. These applications indicate that this family can provide better fits than other well-known classes of distributions. The beauty and importance of the proposed family lies in its ability to model different types of real data.     

A Sarmanov family with beta and gamma marginal distributions: an application to the Bayes premium in a collective risk model

In this paper we firstly develop a Sarmanov–Lee bivariate family of distributions with the beta and gamma as marginal distributions. We obtain the linear correlation coefficient showing that, although it is not a strong family of correlation, it can be greater than the value of this coefficient in the Farlie–Gumbel–Morgenstern family. We also determine other measures for this family: the coefficient of median concordance and the relative entropy, which are analyzed by comparison with the case of independence. Secondly, we consider the problem of premium calculation in a Poisson–Lindley and exponential collective risk model, where the Sarmanov–Lee family is used as a structure function. We determine the collective and Bayes premiums whose values are analyzed when independence and dependence between the risk profiles are considered, obtaining that notable variations in premiums values are obtained even when low levels of correlation are considered.     

A new generalized odd log-logistic family of distributions

We introduce and study general mathematical properties of a new generator of continuous distributions with three extra parameters called the new generalized odd log-logistic family of distributions. The proposed family contains several important classes discussed in the literature as submodels such as the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, entropy measures, and order statistics, which hold for any baseline model, are presented. We also present certain characterization of the proposed distribution and derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behavior of the maximum likelihood estimator via simulation. The importance of the new family is illustrated by means of two real data sets. These applications indicate that the new family can provide better fits than other well-known classes of distributions. The beauty and importance of the new family lies in its ability to model real data.     

Odd-Burr generalized family of distributions with some applications

We study a new family of continuous distributions with two extra shape parameters called the Burr generalized family of distributions. We investigate the shapes of the density and hazard rate function. We derive explicit expressions for some of its mathematical quantities. The estimation of the model parameters is performed by maximum likelihood. We prove the flexibility of the new family by means of applications to two real data sets. Furthermore, we propose a new extended regression model based on the logarithm of the Burr generalized distribution. This model can be very useful to the analysis of real data and provide more realistic fits than other special regression models.     

Some bounds related to Harris family of distributions

As a lifetime distribution, Harris family of distributions are applied to the lifetime of a series system with random number of components. In this paper, properties of various ageing classes of mixtures of Harris family of distributions, where the tilt parameter of a Harris distribution is taken as a random variable, are studied. We obtain an upper bound for maximum error in evaluating its reliability function. Two bounds are also presented for survival function and expectation of the mixed Harris family. We also provide some interesting bounds for its residual survival function. Our results generalize several previous findings in this connection. Some illustrative examples are also provided.     

Topp–Leone generated family of distributions: Properties and applications

We introduce a new family of distributions based on a one-parameter distribution exhibiting bathtub-shaped hazard rates. We study the mathematical properties of the family and estimate its parameters by the method of maximum likelihood. Finally, the usefulness of the family is illustrated using a real dataset.     

The generalized transmuted-G family of distributions

We introduce a new class of continuous distributions called the generalized transmuted-G family which extends the transmuted-G class. We provide six special models of the new family. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The flexibility of the proposed family is illustrated by means of three applications to real data sets.     

Bias bound for the minimax estimator

The bias bound function of an estimator is an important quantity in order to perform globally robust inference. We show how to evaluate the exact bias bound for the minimax estimator of the location parameter for a wide class of unimodal symmetric location and scale family. We show, by an example, how to obtain an upper bound of the bias bound for a unimodal asymmetric location and scale family. We provide the exact bias bound of the minimum distance/disparity estimators under a contamination neighborhood generated from the same distance.     

A package of four edited books on structural equations modeling

We consider two analytical and a bootstrap bias correction scheme existing in the literature for maximum likelihood estimators (MLEs) in the special case of a particular biparametric exponential family, the estimators being obtained from i.i.d. samples. We assess the performances of the estimators through numerical simulations for three particular cases of the family explored here. We observe that the two analytical proposals display very similar behavior for these distributions and that all proposed estimators are effective in reducing bias and mean square error of the MLEs.     

Automatic selection of the proper family for simultaneous confidence intervals

Statisticians often employ simultaneous confidence intervals to reduce the likelihood of their drawing false conclusions when they must make a number of comparisons. To do this properly, it is necessary to consider the family of comparisons over which simultaneous confidence must be assured. Sometimes it is not clear what family of comparisons is appropriate. We describe how computer software can monitor the types of contrasts a user examines, and select the smallest family of contrasts that is likely to be of interest. We also describe how to calculate simultaneous confidence intervals for these families using a hybrid of the Bonferroni and Scheffé methods. Our method is especially suitable for problems with discrete and continuous predictors.     

A new class of slash-elliptical distributions

In this paper, we consider a generalization of the modified slash distribution. We define the new family through the quotient between an elliptically distributed random variable and the power of an exponential random variable with parameter equals to 2, both independent. We use the same idea to extend the model for the multivariate case and study general important properties from the resultant family. We perform inference by the method of moments and maximum likelihood. We present a simulation study which indicates satisfactory parameter recovery by using the estimation approaches. Illustrations reveals that it has potential for doing well in real problems.     

Mixtures of modified t-factor analyzers for model-based clustering,classification, and discriminant analysis

A novel family of mixture models is introduced based on modified t-factor analyzers. Modified factor analyzers were recently introduced within the Gaussian context and our work presents a more flexible and robust alternative. We introduce a family of mixtures of modified t-factor analyzers that uses this generalized version of the factor analysis covariance structure. We apply this family within three paradigms: model-based clustering; model-based classification; and model-based discriminant analysis. In addition, we apply the recently published Gaussian analogue to this family under the model-based classification and discriminant analysis paradigms for the first time. Parameter estimation is carried out within the alternating expectation-conditional maximization framework and the Bayesian information criterion is used for model selection. Two real data sets are used to compare our approach to other popular model-based approaches; in these comparisons, the chosen mixtures of modified t-factor analyzers model performs favourably. We conclude with a summary and suggestions for future work.     

The generalized odd half-Cauchy family of distributions: Properties and applications

We introduce and study general mathematical properties of a new generator of continuous distributions with one extra parameter called the generalized odd half-Cauchy family. We present some special models and investigate the asymptotics and shapes. The new density function can be expressed as a linear mixture of exponentiated densities based on the same baseline distribution. We derive a power series for the quantile function. We discuss the estimation of the model parameters by maximum likelihood and prove empirically the flexibility of the new family by means of two real data sets.     

Manova with covariates and an application in profile analysis

Replacing one of the two marginal distributions in a bivariate normal by a family of symmetrical distributions, we obtain a new family of symmetric bivariate distributions. We use the Tiku - Suresh (1990) method to estimate the parameters of this new bivariate family. We define a Hotelling - type statistic to test the mean vector and evaluate the asymptotic power of this statistic relative to the Hotelling T² statistic. We show that the former is considerably more powerful.     

The exponentiated transmuted-G family of distributions: Theory and applications

In this paper, a new family of continuous distributions called the exponentiated transmuted-G family is proposed which extends the transmuted-G family defined by Shaw and Buckley (2007). Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, and order statistics are derived. Some special models of the new family are provided. The maximum likelihood is used for estimating the model parameters. We provide the simulation results to assess the performance of the proposed model. The usefulness and flexibility of the new family is illustrated using real data.     

The type I half-logistic family of distributions

We study general mathematical properties of a new class of continuous distributions with an extra positive parameter called the type I half-logistic family. We present some special models and investigate the asymptotics and shapes. The new density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. We derive a power series for the quantile function. Explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon and Rényi entropies and order statistics are determined. We introduce a bivariate extension of the new family. We discuss the estimation of the model parameters by maximum likelihood and illustrate its potentiality by means of two applications to real data.     

The Exponentiated G Poisson Model

We study a new family of distributions defined by the minimum of the Poisson random number of independent identically distributed random variables having a general exponentiated G distribution. Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics and their moments, reliability, and Shannon entropy are derived. Maximum likelihood estimation of the model parameters is investigated. Two special models of the new family are discussed. We perform an application to a real data set to show the potentiality of the proposed family.