Fundamental Equations of BLUE and BLUP in the Multivariate Linear Model with Applications

Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), and best linear unbiased predictor (BLUP) in the general linear model with new observations are generalized to the general multivariate linear model. The fundamental equations of BLUE and BLUP in the multivariate linear model are derived by two methods, including the vectorization method and projection method. By using the matrix rank method, some new results of linear BLUE-sufficiency, linear BLUP-sufficiency, and the equality of OLSE, BLUE, and BLUP are given in the multivariate linear model.     

Sufficient and admissible estimators in general multivariate linear model

The notion of linear sufficiency in general Gauss–Markov model is extended to a general multivariate linear model for any specific set of estimable functions. A general formula of the difference between the dispersion matrix of the BLUE in the original model and that in the transformed model is provided, which brings some further contributions to the theory of linear sufficiency. Moreover, a general formula of the change of BLUE due to transformation is obtained. The analysis here leads to some results, some of which are known in the literature. Besides linear sufficiency, the admissibility of a linear statistic is also extended to the multivariate case.     

Bayesian quantile regression for skew-normal linear mixed models

Linear mixed models have been widely used to analyze repeated measures data which arise in many studies. In most applications, it is assumed that both the random effects and the within-subjects errors are normally distributed. This can be extremely restrictive, obscuring important features of within-and among-subject variations. Here, quantile regression in the Bayesian framework for the linear mixed models is described to carry out the robust inferences. We also relax the normality assumption for the random effects by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in the linear mixed models. For posterior inference, we propose a Gibbs sampling algorithm based on a mixture representation of the asymmetric Laplace distribution and multivariate skew-normal distribution. The procedures are demonstrated by both simulated and real data examples.     

Pitman closeness,monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer than the BLIE. Next, we introduce the notions of Pitman monotonicity and Pitman consistency, and then establish that both BLUE and BLIE possess these two properties.     

The admissible minimax estimator in Gauss–Markov model under a balanced loss function

In this paper, we investigate the admissible minimax estimator (AME) of regression coefficient in Gauss–Markov model under a balanced loss function. In the class of homogeneous linear estimators, we obtain the AME under two occasions, respectively. We also prove that the AME is a shrinkage estimator of the best linear unbiased estimator (BLUE). Furthermore, we prove that the AME dominates the BLUE under certain conditions.     

Local influence analysis for regression models with scale mixtures of skew-normal distributions

The robust estimation and the local influence analysis for linear regression models with scale mixtures of multivariate skew-normal distributions have been developed in this article. The main virtue of considering the linear regression model under the class of scale mixtures of skew-normal distributions is that they have a nice hierarchical representation which allows an easy implementation of inference. Inspired by the expectation maximization algorithm, we have developed a local influence analysis based on the conditional expectation of the complete-data log-likelihood function, which is a measurement invariant under reparametrizations. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex and with Cook's well-known approach it can be very difficult to obtain measures of the local influence. Some useful perturbation schemes are discussed. In order to examine the robust aspect of this flexible class against outlying and influential observations, some simulation studies have also been presented. Finally, a real data set has been analyzed, illustrating the usefulness of the proposed methodology.     

Some matrix results related to a partitioned singular linear model

Consider the linear model (y, Xβ V), where the model matrix X may not have a full column rank and V might be singular. In this paper we introduce a formula for the difference between the BLUES of Xβ under the full model and the model where one observation has been deleted. We also consider the partitioned linear regression model where the model matrix is (X₁: X₂) the corresponding vector of unknown parameters being (β′₁ : β′₂)′. We show that the BLUE of X₁ β₁ under a specific reduced model equals the corresponding BLUE under the original full model and consider some interesting consequences of this result.     

Exact and Approximate Statistical Inference for Nonlinear Regression and the Estimating Equation Approach

The exact density distribution of the non‐linear least squares estimator in the one‐parameter regression model is derived in closed form and expressed through the cumulative distribution function of the standard normal variable. Several proposals to generalize this result are discussed. The exact density is extended to the estimating equation (EE) approach and the non‐linear regression with an arbitrary number of linear parameters and one intrinsically non‐linear parameter. For a very special non‐linear regression model, the derived density coincides with the distribution of the ratio of two normally distributed random variables previously obtained by Fieler almost a century ago, unlike other approximations previously suggested by other authors. Approximations to the density of the EE estimators are discussed in the multivariate case. Numerical complications associated with the non‐linear least squares are illustrated, such as non‐existence and/or multiple solutions, as major factors contributing to poor density approximation. The non‐linear Markov–Gauss theorem is formulated on the basis of the near exact EE density approximation.     

A note on the Cook''s distance

A modification of the classical Cook's distance is proposed, providing us with a generalized Mahalanobis distance in the context of multivariate elliptical linear regression models. We establish the exact distribution of a pivotal type statistics based on this generalized Mahalanobis distance, which provides critical points for the identification of outlier data points. Based on the equivalence between the modified Cook's distance and what is called the mean-shift multivariate outlier elliptical model, twelve new modifications are proposed for the Cook's distance. We also describe the explicit relationship between the Cook's distance and the likelihood displacement with the modified Cook's distance. We illustrate the procedure with some examples, in the context of multiple and multivariate linear regression.     

Multivariate autoregressive extreme value process and its application for modeling the time series properties of the extreme daily asset prices

Abstract

In this article we suggest a new multivariate autoregressive process for modeling time-dependent extreme value distributed observations. The idea behind the approach is to transform the original observations to latent variables that are univariate normally distributed. Then the vector autoregressive DCC model is fitted to the multivariate latent process. The distributional properties of the suggested model are extensively studied. The process parameters are estimated by applying a two-stage estimation procedure. We derive a prediction interval for future values of the suggested process. The results are applied in an empirically study by modeling the behavior of extreme daily stock prices.     

Detection of multiple change-points in multivariate data

The statistical analysis of change-point detection and estimation has received much attention recently. A time point such that observations follow a certain statistical distribution up to that point and a different distribution – commonly of the same functional form but different parameters after that point – is called a change-point. Multiple change-point problems arise when we have more than one change-point. This paper develops a method for multivariate normally distributed data to detect change-points and estimate within-segment parameters using maximum likelihood estimation.     

A new class of survival regression models with heavy-tailed errors: robustness and diagnostics

Birnbaum-Saunders models have largely been applied in material fatigue studies and reliability analyses to relate the total time until failure with some type of cumulative damage. In many problems related to the medical field, such as chronic cardiac diseases and different types of cancer, a cumulative damage caused by several risk factors might cause some degradation that leads to a fatigue process. In these cases, BS models can be suitable for describing the propagation lifetime. However, since the cumulative damage is assumed to be normally distributed in the BS distribution, the parameter estimates from this model can be sensitive to outlying observations. In order to attenuate this influence, we present in this paper BS models, in which a Student-t distribution is assumed to explain the cumulative damage. In particular, we show that the maximum likelihood estimates of the Student-t log-BS models attribute smaller weights to outlying observations, which produce robust parameter estimates. Also, some inferential results are presented. In addition, based on local influence and deviance component and martingale-type residuals, a diagnostics analysis is derived. Finally, a motivating example from the medical field is analyzed using log-BS regression models. Since the parameter estimates appear to be very sensitive to outlying and influential observations, the Student-t log-BS regression model should attenuate such influences. The model checking methodologies developed in this paper are used to compare the fitted models.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Bayesian assessment of assumptions of regression analysis

Many statistical procedures are based on the models which specify the conditions under which the data are generated. Many applications of linear regression, for example, assume that:(i) the observations are independent; (ii) the errors in the observations are identically distributed; (iii) each error has a normal distribution with mean zero and unknown variance σ²> 0. Previous works have examined individual departures from these assumptions. Here we examine composite departures. It is assumed that the error distribution in a linear model is power-exponential and that the observations are generated via a first order autoregressive model with the possibility of spurious observations. The consequences are illustrated via an example.     

Correlated random effects regression analysis for a log-normally distributed variable

In regression analysis, it is assumed that the response (or dependent variable) distribution is Normal, and errors are homoscedastic and uncorrelated. However, in practice, these assumptions are rarely satisfied by a real data set. To stabilize the heteroscedastic response variance, generally, log-transformation is suggested. Consequently, the response variable distribution approaches nearer to the Normal distribution. As a result, the model fit of the data is improved. Practically, a proper (seems to be suitable) transformation may not always stabilize the variance, and the response distribution may not reduce to Normal distribution. The present article assumes that the response distribution is log-normal with compound autocorrelated errors. Under these situations, estimation and testing of hypotheses regarding regression parameters have been derived. From a set of reduced data, we have derived the best linear unbiased estimators of all the regression coefficients, except the intercept which is often unimportant in practice. Unknown correlation parameters have been estimated. In this connection, we have derived a test rule for testing any set of linear hypotheses of the unknown regression coefficients. In addition, we have developed the confidence ellipsoids of a set of estimable functions of regression coefficients. For the fitted regression equation, an index of fit has been proposed. A simulated study illustrates the results derived in this report.     

A note on Stein's lemma for multivariate elliptical distributions

When two random variables are bivariate normally distributed Stein's original lemma allows to conveniently express the covariance of the first variable with a function of the second. Landsman and Neslehova (2008) extend this seminal result to the family of multivariate elliptical distributions. In this paper we use the technique of conditioning to provide a more elegant proof for their result. In doing so, we also present a new proof for the classical linear regression result that holds for the elliptical family.     

Multivariate Linear Regression Model with Elliptically Contoured Distributed Errors and Monotone Missing Dependent Variables

In this article, the multivariate linear regression model is studied under the assumptions that the error term of this model is described by the elliptically contoured distribution and the observations on the response variables are of a monotone missing pattern. It is primarily concerned with estimation of the model parameters, as well as with the development of the likelihood ratio test in order to examine the existence of linear constraints on the regression coefficients. An illustrative example is presented for the explanation of the results.     

A note on predictive inference for multivariate elliptically contoured distributions

The prediction problem is considered for the multivariate regression model with an elliptically contoured error distribution. We show that the predictive distribution under elliptical errors assumption is the same as that obtained under normally distributed error in both the Bayesian approach using an im-proper prior and the classical approach. This gives inference robustness with respect to departures from the reference case of independent sampling from the normal distribution.     

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.