Multivariate Birnbaum–Saunders regression model

In this paper, we propose a multivariate log-linear Birnbaum–Saunders regression model. We discuss maximum-likelihood estimation of the model parameters and provide closed-form expressions for the score function and for Fisher's information matrix. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum-likelihood estimator. Some influence methods, such as the local influence and generalized leverage are discussed and the normal curvatures for studying local influence are derived under some perturbation schemes. Further, a test for the homogeneity of the shape parameter of the multivariate regression model is investigated. A real data set is presented for illustrative purposes.     

Principal components regression estimator and a test for the restrictions

In this article, we introduce restricted principal components regression (RPCR) estimator by combining the approaches followed in obtaining the restricted least squares estimator and the principal components regression estimator. The performance of the RPCR estimator with respect to the matrix and the generalized mean square error are examined. We also suggest a testing procedure for linear restrictions in principal components regression by using singly and doubly non-central F distribution.     

On a characterization of the dispersion matrix based on the properties of regression

This paper provides a partial solution to a problem posed by J. Neyman (1965) regarding the characterization of multivariate negative binomial distribution based on the properties of regression. It is shown that some of the properties of regression characterize the form of the nonsingular dispersion matrix of the parent distribution, which, interestingly enough, corresponds to only two types viz. those of positive and negative multivariate binomial distributions.     

Regression analysis of autocorrelated poisson-distribution) data

A regression model assuming Poisson-dia distributed data. with autocorrelated errors falls into the class of regression models that; have the error structure which is both heteroscedastic and autocorrelated. In general, this class of regression models are not estimable. However, due to the properties of the Poisson distribution that the variance is equal to the mean, this regression model on Poisson-distributed data with autocorrelated. errors is estimable. In this note the special structure of the covarlance matrix of the model with the first order auto-correlated error Is derived utilizing this property, A method based on the least squares method of Frome, Kutner, and Beauchamp (1973), supplemented by steps for handling autocorrelation in studies of time series analysis, nonlinear regression, and econometrics is presented for obtaining generalized least squares estimates for the parameters of the model.     

Testing for a change lit the regression matrix of a multivariate linear model

A Bayesian test procedure Is developed to test; the null hypothesis of no change In the regression matrix of a multivariate lin¬ear model against the alternative hypothesis of exactly one change The resulting test is based on the marginal posterior distribution of the change point; To illustrate the test procedure a numerical example using a bivariate regression model is considered.     

Structural changes in multivariate regression models

This study generalizes the work of chin choy and Broemeling (1980) who investigated the change in the regression parameters of univariate linear models.

The marginal posterior distributions of the change point, the regression matrices,and the precision matrix are found with the use of a proper multivariate normal-Wishart distribution for the parameters of the model.

A numerical study is undertaken in order to gain some insight into the effect that changes in sample size and certain parameter values have on these marginal posterior distributions.     

Fast regression surrogates for computer models with time-dependent outputs

The study of physical processes is often aided by computer models or codes. Computer models that simulate such processes are sometimes computationally intensive and therefore not very efficient exploratory tools. In this paper, we address computer models characterized by temporal dynamics and propose new statistical correlation structures aimed at modelling their time dependence. These correlations are embedded in regression models with input-dependent design matrix and input-correlated errors that act as fast statistical surrogates for the computationally intensive dynamical codes. The methods are illustrated with an automotive industry application involving a road load data acquisition computer model.     

Corrected maximum likelihood estimators in heteroscedastic symmetric nonlinear models

In this article, we derive general matrix formulae for second-order biases of maximum likelihood estimators (MLEs) in a class of heteroscedastic symmetric nonlinear regression models, thus generalizing some results in the literature. This class of regression models includes all symmetric continuous distributions, and has a wide range of practical applications in various fields such as engineering, biology, medicine and economics, among others. The variety of distributions with different kurtosis coefficients than the normal may give more flexibility in the choice of an appropriate distribution, particularly to accommodate outlying and influential observations. We derive a joint iterative process for estimating the mean and dispersion parameters. We also present simulation studies for the biases of the MLEs.     

A simple approximation of the sampling distribution of least absolute residuals regression estimates

For multivariate regression with a symmetric disturbance distribution, the error in the least absolute residuals estimator is approximately multivariate normally distributed with mean zero and variance matrix λ²(X′X)^?1, where X is the matrix of K explanatory variables and T observations, and λ ²/T is the variance of the median of a sample of size T from the disturbance distribution. The approximate sampling theory is validated by extensive Monte Carlo studies, and some directions of possible refinement emerge.     

Multivariate log-Birnbaum–Saunders regression models

In this paper, we present a multivariate version of the skewed log-Birnbaum–Saunders regression model. This new family of distributions holds good properties such as marginal variables following univariate skewed log-Birnbaum–Saunders distributions, besides presenting the usual log-Birnbaum–Saunders distribution as a particular case. Furthermore, the model parameters are estimated through maximum-likelihood methods, a closed-form expression for the Fisher’s information matrix is presented, and testing hypothesis for model parameters is performed. Two real datasets are analyzed and results are discussed.     

Further identities for the Wishart distribution with applications in regression

Matrix analogues are given for a known scalar identity which relates certain expectations with respect to the Wishart distribution. (The scalar identity was independently derived by C. Stein and L. Haff.) The matrix analogues are more aptly called “matrix extensions.” They can be derived by using the scalar identity; nevertheless, they are seen (in quite elementary terms) to be more general than the latter. A method of doing multivariate calculations is developed from the identities, and several examples are worked in detail. We compute the first two moments of the regression coefficients and another matrix arising in regression analysis. Also, we give a new result for the matrix analogue of squared multiple correlation: the bias correction of Ezekiel (1930), a result often used in model building, is extended to the case of two or more dependent variables.     

An ECM Estimation Approach for Analyzing Multivariate Skew-Normal Data with Dropout

In this article, an ECM algorithm is developed to obtain the maximum likelihood estimates of parameters where multivariate skew-normal distribution is used for analyzing longitudinal skewed normal regression data with dropout. A simulation study is performed to investigate the performance of the presented algorithm. Also, the methodology is illustrated through two applications and the results of proposed methodology are compared with ECM under multivariate normal assumption using AIC and BIC criteria. Standard errors of parameter estimates are obtained by asymptotic observed information matrix.     

The McDonald Weibull model

For the first time, we propose a five-parameter lifetime model called the McDonald Weibull distribution to extend the Weibull, exponentiated Weibull, beta Weibull and Kumaraswamy Weibull distributions, among several other models. We obtain explicit expressions for the ordinary moments, quantile and generating functions, mean deviations and moments of the order statistics. We use the method of maximum likelihood to fit the new distribution and determine the observed information matrix. We define the log-McDonald Weibull regression model for censored data. The potentiality of the new model is illustrated by means of two real data sets.     

Bias corrected estimates in multivariate student t regression models

The t distribution has proved to be a useful alternative to the normal distribution especially When robust estimation is desired. We consider the multivariate nonlinear Student-t regression model and show that the biased of the estimates of the regression coefficients can be computed from an auxiliary generalized linear regression. We give a formula for the biases of the estimates of the parameters in the scale matrix, which also can be computed by means of a generalized linear regression. We briefly discuss some important special cases and present simulation results which indicate that our bias-corrected estimates outperform the uncorrected ones in small samples.     

Adjusted Pearson residuals in beta regression models

In this paper, matrix formulae of order n^?1, where n is the sample size, for the first two moments of Pearson residuals are obtained in beta regression models. Adjusted Pearson residuals are also obtained, having, to this order, expected value zero and variance one. Monte Carlo simulation results are presented illustrating the behaviour of both adjusted and unadjusted residuals.     

A note on classical Stein-type estimators in elliptically contoured models

In this paper the conditions under which a broad class of Stein-type estimators dominates the best invariant unbiased estimator of the mean of an elliptically contoured population have been established. The superiority conditions are derived for both known and unknown scale structures. Also an example is given when the general scale matrix is assumed to be known in linear regression.     

The Density Functions of the Singular Quaternion Normal Matrix and the Singular Quaternion Wishart Matrix

In this article, we give the density functions of the singular quaternion normal matrix and the singular quaternion Wishart matrix. Furthermore, we also give the density functions of certain singular quaternion β-matrix and the singular quaternion F-matrix in terms of the density function of the singular quaternion Wishart matrix and hypergeometric functions of quaternion matrix argument.     

Added variable plots for linear regression with censored data

In linear regression the structure of the hat matrix plays an important part in regression diagnostics. In this note we investigate the properties of the hat matrix for regression with censored responses in the presence of one or more explanatory variables observed without censoring. The censored points in the scatterplot are renovated to positions had they been observed without censoring in a renovation process based on Buckley-James censored regression estimators. This allows natural links to be established with the structure of ordinary least squares estimators. In particular, we show that the renovated hat matrix may be partitioned in a manner which assists in deciding whether further explanatory variables should be added to the linear model. The added variable plot for regression with censored data is developed as a diagnostic tool for this decision process.     

Optimal sample size allocation for multi-level stress testing with Weibull regression under Type-II censoring

We discuss the optimal allocation problem in a multi-level stress test with Type-II censoring and Weibull (extreme value) regression model. We derive the maximum-likelihood estimators and their asymptotic variance–covariance matrix through the Fisher information. Four optimality criteria are used to discuss the optimal allocation problem. Optimal allocation of units, both exactly for small sample sizes and asymptotically for large sample sizes, for two- and four-stress-level situations are determined numerically. Conclusions and discussions are provided based on the numerical studies.     

Diagonalization matrix and its application in distribution theory

Some matrix representations of diverse diagonal arrays are studied in this work; the results allow new definitions of classes of elliptical distributions indexed by kernels mixing Hadamard and usual products. A number of applications are derived in the setting of prior densities from the Bayesian multivariate regression model and families of non-elliptical distributions, such as the matrix multivariate generalized Birnbaum–Saunders density. The philosophy of the research about matrix representations of quadratic and inverse quadratic forms can be extended as a methodology for exploring possible new applications in non-standard distributions, matrix transformations and inference.