A class of rectangle-screened multivariate normal distributions and its applications

A screening problem is tackled by proposing a parametric class of distributions designed to match the behavior of the partially observed screened data. This class is obtained from the nontruncated marginal of the rectangle-truncated multivariate normal distributions. Motivations for the screened distribution as well as some of the basic properties, such as its characteristic function, are presented. These allow us a detailed exploration of other important properties that include closure property in linear transformation, in marginal and conditional operations, and in a mixture operation as well as the first two moments and some sampling distributions. Various applications of these results to the statistical modelling and data analysis are also provided.     

A continuous general multivariate distribution and its properties

Starting from two known continuous univariate distributions, a bivariate distribution is constructed depending on a parameter which measures the degree of stochastic dependence between the two random variables. From the foregoing construction we then pass to a multivariate-type distribution, constructed using only univariate distributions and an association matrix. Some properties of the multivariate and bivariate case are studied.     

A class of weighted normal distributions and its variants useful for inequality constrained analysis

This article develops a class of the weighted normal distributions for which the probability density function has the form of a product of a normal density and a weight function. The class constitutes marginal distributions obtained from various kinds of doubly truncated bivariate normal distributions. This class of distributions strictly includes the normal, skew–normal and two-piece skew–normal and is useful for selection modelling and inequality constrained normal mean analysis. Some distributional properties and Bayesian perspectives of the class are given. Probabilistic representation of the distributions is also given. The representation is shown to be straightforward to specify distribution and to implement computation, with output readily adapted for required analysis. Necessary theories and illustrative examples are provided.     

Estimation of the parameters of a regression model with a multivariate t error variable

This paper deals with a regression model for several vari¬ables under the assumption that the errors have a multivariate t-distribution. The parameters of the model, the regression parameters, as well as the scale parameters and the degress of freedom of the error variable are estimated and the estimation procedure is illustrated by a numerical example, Also, the prop¬erties of the estimators and tests for the regression parameters are discussed.     

Tests of heteroscedasticity and correlation in multivariate t regression models with AR and ARMA errors

Heteroscedasticity checking in regression analysis plays an important role in modelling. It is of great interest when random errors are correlated, including autocorrelated and partial autocorrelated errors. In this paper, we consider multivariate t linear regression models, and construct the score test for the case of AR(1) errors, and ARMA(s,d) errors. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied. Based on modified profile likelihood, the adjusted score test is also developed. The finite sample performance of the tests is investigated through Monte Carlo simulations, and also the tests are illustrated with two real data sets.     

Modified likelihood ratio tests in heteroskedastic multivariate regression models with measurement error

In this paper, we develop modified versions of the likelihood ratio test for multivariate heteroskedastic errors-in-variables regression models. The error terms are allowed to follow a multivariate distribution in the elliptical class of distributions, which has the normal distribution as a special case. We derive the Skovgaard-adjusted likelihood ratio statistics, which follow a chi-squared distribution with a high degree of accuracy. We conduct a simulation study and show that the proposed tests display superior finite sample behaviour as compared to the standard likelihood ratio test. We illustrate the usefulness of our results in applied settings using a data set from the WHO MONICA Project on cardiovascular disease.     

A new class of skew distributions with climate data analysis

In this paper, we develop a new general class of skew distributions with flexibility properties on the tails. Moreover, such class can provide heavy and light tails. Some of its mathematical properties are studied, including the quantile function, the moments, the moment generating function and the mean of deviations. New skew distributions are derived and used to construct new models capturing asymmetry inherent to data. The estimation of the class parameters is investigated by the method of maximum likelihood and the performance of the estimators is assessed by a simulation study. Applications of the proposed distribution are explored for two climate data sets. The first data set concerns the annual heat wave index and the second data set involves temperature and precipitation measures from the meteorological station located at Schiphol, Netherlands. Data fitting results show that our models perform better than the competitors.     

Robust variable selection in finite mixture of regression models using the t distribution

Abstract

Variable selection in finite mixture of regression (FMR) models is frequently used in statistical modeling. The majority of applications of variable selection in FMR models use a normal distribution for regression error. Such assumptions are unsuitable for a set of data containing a group or groups of observations with heavy tails and outliers. In this paper, we introduce a robust variable selection procedure for FMR models using the t distribution. With appropriate selection of the tuning parameters, the consistency and the oracle property of the regularized estimators are established. To estimate the parameters of the model, we develop an EM algorithm for numerical computations and a method for selecting tuning parameters adaptively. The parameter estimation performance of the proposed model is evaluated through simulation studies. The application of the proposed model is illustrated by analyzing a real data set.     

A Note on parameter and standard error estimation in adaptive robust regression

This paper introduces practical methods of parameter and standard error estimation for adaptive robust regression where errors are assumed to be from a normal/independent family of distributions. In particular, generalized EM algorithms (GEM) are considered for the two cases of t and slash families of distributions. For the t family, a one step method is proposed to estimate the degree of freedom parameter. Use of empirical information is suggested for standard error estimation. It is shown that this choice leads to standard errors that can be obtained as a by-product of the GEM algorithm. The proposed methods, as discussed, can be implemented in most available nonlinear regression programs. Details of implementation in SAS NLIN are given using two specific examples.     

On the bias and mean square error of the least square estimator in a regression model with two inequality constraints and multivariate t error terms

We derive and numerically evaluate the bias and mean square error of the inequality constrained least squares estimator in a model with two inequality constraints and multivariate terror terms. Our results suggest that qualitatively, the estimator properties found for models with normal errors carry over to the case of multivariate terrors.     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.     

A new regression model for bimodal data and applications in agriculture

We define the odd log-logistic exponential Gaussian regression with two systematic components, which extends the heteroscedastic Gaussian regression and it is suitable for bimodal data quite common in the agriculture area. We estimate the parameters by the method of maximum likelihood. Some simulations indicate that the maximum-likelihood estimators are accurate. The model assumptions are checked through case deletion and quantile residuals. The usefulness of the new regression model is illustrated by means of three real data sets in different areas of agriculture, where the data present bimodality.     

A new log-location regression model: estimation,influence diagnostics and residual analysis

We introduce a log-linear regression model based on the odd log-logistic generalized half-normal distribution [7 G.M. Cordeiro, M. Alizadeh, R.R. Pescim, and E.M.M. Ortega, The odd log-logistic generalized half-normal lifetime distribution: Properties and applications, Comm. Statist. Theory Methods (2015), accepted for publication. [Google Scholar]]. Some of its structural properties including explicit expressions for the density function, quantile and generating functions and ordinary moments are derived. We estimate the model parameters by the maximum likelihood method. For different parameter settings, proportion of censoring and sample size, some simulations are performed to investigate the behavior of the estimators. We derive the appropriate matrices for assessing local influence diagnostics on the parameter estimates under different perturbation schemes. We also define the martingale and modified deviance residuals to detect outliers and evaluate the model assumptions. In addition, we demonstrate that the extended regression model can be very useful in the analysis of real data and provide more realistic fits than other special regression models. The potentiality of the new regression model is illustrated by means of a real data set.     

Survival analysis for a new compounded bivariate failure time distribution in shock and competing risk models via an EM algorithm

Abstract

In this paper, two bivariate models based on the proposed methods of Marshall and Olkin are introduced. In the first model, the new bivariate distribution is presented based on the proposed method of Marshall and Olkin (1967 Marshall, A. W., and I. Olkin. 1967. A multivariate exponential distribution. Journal of the American Statistical Association 62 (317):30–44. doi: 10.2307/2282907.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) which has natural interpretations, and it can be applied in fatal shock models or in competing risks models. In the second model, the proposed method of Marshall and Olkin (1997 Marshall, A. W., and I. Olkin. 1997. A new method of adding a parameter to a family of distributions with application to the exponential and weibull families. Biometrika 84 (3):641–52. doi: 10.1093/biomet/84.3.641.[Crossref], [Web of Science ®] , [Google Scholar]) is generalized to bivariate case and a new bivariate distribution is introduced. We call these new distributions as the bivariate Gompertz (BGP) distribution and bivariate Gompertz-geometric (BGPG) distribution, respectively. Moreover, the BGP model can be obtained as a special case of the BGPG model. Then, we present various properties of the new bivariate models. In this regard, the joint and conditional density functions, the joint cumulative distribution function can be obtained in compact forms. Also, the aging properties and the bivariate hazard gradient are discussed. This model has five unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters, and it is computationally quite tractable. Also, Monte Carlo simulations are performed to investigate the effectiveness of the proposed algorithm. Finally, we analyze three real data sets for illustrative purposes.     

EM algorithm-based likelihood estimation for a generalized Gompertz regression model in presence of survival data with long-term survivors: an application to uterine cervical cancer data

In this paper we develop a regression model for survival data in the presence of long-term survivors based on the generalized Gompertz distribution introduced by El-Gohary et al. [The generalized Gompertz distribution. Appl Math Model. 2013;37:13–24] in a defective version. This model includes as special case the Gompertz cure rate model proposed by Gieser et al. [Modelling cure rates using the Gompertz model with covariate information. Stat Med. 1998;17:831–839]. Next, an expectation maximization algorithm is then developed for determining the maximum likelihood estimates (MLEs) of the parameters of the model. In addition, we discuss the construction of confidence intervals for the parameters using the asymptotic distributions of the MLEs and the parametric bootstrap method, and assess their performance through a Monte Carlo simulation study. Finally, the proposed methodology was applied to a database on uterine cervical cancer.     

Optimal tolerance regions for future regression vector and residual sum of squares of multiple regression model with multivariate spherically contoured errors

This paper considers multiple regression model with multivariate spherically symmetric errors to determine optimal β-expectation tolerance regions for the future regression vector (FRV) and future residual sum of squares (FRSS) by using the prediction distributions of some appropriate functions of future responses. The prediction distribution of the FRV, conditional on the observed responses, is multivariate Student-t distribution. Similarly, the prediction distribution of the FRSS is a beta distribution. The optimal β-expectation tolerance regions for the FRV and FRSS have been obtained based on the F -distribution and beta distribution, respectively. The results in this paper are applicable for multiple regression model with normal and Student-t errors.