RECURSIVE RESIDUALS FOR MULTIVARIATE REGRESSION MODELS

Recursive residuals and their relationship to the recursive estimation of regression parameters have been developed for unvaried regression mod els. Such residuals and estimates have been used to test the constancy of regression over time. The current paper extends this work to multivariate regression modal.     

EXACT DISTRIBUTION OF CERTAIN GENERAL TEST STATISTICS IN MULTIVARIATE ANALYSIS

In this paper, exact solution of Wilks' type-B integral equation has been obtained in its most general form as a series of weighted gamma distributions. This general result then gives the distributions of many test statistics in multivariate analysis. In particular the distributions of Wilks' Λ, the sphericity test criterion, and Bartlett's test statistic, are derived in easily computable form.     

ON CHARACTERIZING DISTRIBUTIONS VIA LINEARITY OF REGRESSION FOR ORDER STATISTICS

Let X₁X_n be a random sample from an absolutely continuous distribution with the corresponding order statistics X_1:n≤X_2:n≤X_n:n. A complete solution of the problem, posed in 1967 by T. Ferguson, of determining the distribution by linearity of regression of X_k+2:n with respect to X_k:n is given. The only possible distributions are of the exponential, power and Pareto type. A linear regression relation for exponents of order statistics is also considered.     

REGRESSION DIAGNOSTIC USING LOCAL INFLUENCE: A REVIEW

The local influence approach of Cook [1] Cook, R. D. 1986. Assessment of Local Influence. Journal Of The Royal Statistical Society Series B-Methodological, 48: 133–169.  [Google Scholar]to regression diagnostic is developed and discussed, and compared with Cook's [2] Cook, R. D. 1977. Detection of Influential Observations in Linear Regression. Technometrics, 19: 15–18. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]deletion approach. The ability of the local influence approach to handle cases simultaneously, as well as some of its theoretical and practical difficulties, are reviewed. The perturbation ideas of the approach are applied to the linear model making distinction between the local perturbations on the assumptions of the model and the data.     

RANDOM PERMUTATION TESTING IN MULTIPLE LINEAR REGRESSION

We developed an alternative random permutation testing method for multiple linear regression, which is an improvement over the existing one proposed by [1] Kennedy, P. E. 1995. Randomization tests in econometrics. Journal of Business and Economic Statistics, 13: 85–94. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] or [2] Freedman, D. and Lane, D. 1983. A nonstochastic interpretation of reported significance levels. Journal of Business and Economic Statistics, 1: 292–298. [Taylor & Francis Online] , [Google Scholar].     

DO NOT WEIGHT FOR HETEROSCEDASTICITY IN NONPARAMETRIC REGRESSION

The potential role of weighting in kernel regression is examined. The concept that weighting has something to do with heteroscedastic errors is shown to be false. However, weighting does affect bias, and ways in which this might be exploited are indicated.     

DETECTING INFLUENTIAL OBSERVATIONS IN SLICED INVERSE REGRESSION ANALYSIS

The detection of influential observations on the estimation of the dimension reduction subspace returned by Sliced Inverse Regression (SIR) is considered. Although there are many measures to detect influential observations in related methods such as multiple linear regression, there has been little development in this area with respect to dimension reduction. One particular influence measure for a version of SIR is examined and it is shown, via simulation and example, how this may be used to detect influential observations in practice.     

SADDLEPOINT METHODS FOR BOOTSTRAP CONFIDENCE BANDS IN NONPARAMETRIC REGRESSION

Bootstrap techniques have been used to construct confidence bands in nonparametric regression problems (Härdle & Bowman, 1988). Yet the required simulation is generally computationally intensive and therefore makes it difficult to conduct further investigations. In this paper, two saddlepoint methods are considered as alternatives to the naive simulation procedure. Some improvements to Härdle & Bowman's bootstrap method are suggested. The improvements are numerically verified using these efficient and accurate analytic methods.     

A MULTIVARIATE NORMAL MODEL FOR PEDIGREE AND LONGITUDINAL DATA AND THE SOFTWARE 'FISHER'

In 1918 R.A. Fisher published an interpretation of covariation between relatives in terms of Mendelian inheritance, which has allowed inference on genetic and environmental components of variation from plant, animal and human pedigree data. Fisher had introduced maximum likelihood six years earlier. His 1918 paper abo contained the basics of linear regression and decomposition of variance. These concepts have now been united to allow flexible modelling of the mean and covariance structure of non-independent data on continuous traits, using maximum likelihood under a multivariate normal assumption. FISHER is a software package, designed for pedigree analysis and easily adapted for repeated measures and longitudinal data analysis. A range of applications illustrate FISHER as a useful statistical tool. Issues related to assumptions, tests-of-fit, and robustness of inference are discussed.     

Multiple deletion measures and conditional influence in regression model

The joint effect of the deletion of the ith and jih cases is given by Gray and Ling (1984), they discussed the influence measures for influential subsets in linear regression analysis. The present paper is concerned with multiple sets of deletion measures in the linear regression model. In particular we are interested in the effects of the jointly and conditional influence analysis for the detection of two influential subsets.     

DISPERSION CONTROL FOR MULTIVARIATE PROCESSES

This paper presents some techniques for monitoring and controlling the dispersion of multivariate normal processes based on subgroup data. The procedures involve use of independent statistics resulting from the decomposition of the covariance matrix. Those that do not depend on prior estimates of the process covariance matrix are particularly attractive to short-run or low volume manufacturing environments.     

Influence diagnostics for Student-t censored linear regression models

In this paper, we extend the censored linear regression model with normal errors to Student-t errors. A simple EM-type algorithm for iteratively computing maximum-likelihood estimates of the parameters is presented. To examine the performance of the proposed model, case-deletion and local influence techniques are developed to show its robust aspect against outlying and influential observations. This is done by the analysis of the sensitivity of the EM estimates under some usual perturbation schemes in the model or data and by inspecting some proposed diagnostic graphics. The efficacy of the method is verified through the analysis of simulated data sets and modelling a real data set first analysed under normal errors. The proposed algorithm and methods are implemented in the R package CensRegMod.     

A CONSISTENT MODEL SPECIFICATION TEST FOR A REGRESSION FUNCTION BASED ON NONPARAMETRIC WAVELET ESTIMATION

In this paper we present a consistent specification test of a parametric regression function against a general nonparametric alternative. The proposed test is based on wavelet estimation and it is shown to have similar rates of convergence to the more commonly used kernel based tests. Monte Carlo simulations show that this test statistic has adequate size and high power and that it compares favorably with its kernel based counterparts in small samples.     

APPROXIMATIONS TO SOME TEST STATISTICS FOR PERMUTATION TESTS IN A COMPLETELY RANDOMIZED DESIGN1

Under a randomization model for a completely randomized design permutation tests are considered based on the usual F statistic and on a multi-response permutation procedure statistic. For the first statistic the first two moments are obtained so a comparision with the distribution under the normal theory model can be made. The second statistic is shown to converge in distribution to an infinite weighted sum of chi-squared variates, the weights being the limits of the eigenvalues of a matrix depending on the distance measure used and the order statistics of the observations.     

SEMIPARAMETRIC ESTIMATION OF THE ERROR DISTRIBUTION IN MULTIVARIATE REGRESSION USING COPULAS

A semiparametric method is developed to estimate the dependence parameter and the joint distribution of the error term in the multivariate linear regression model. The nonparametric part of the method treats the marginal distributions of the error term as unknown, and estimates them using suitable empirical distribution functions. Then the dependence parameter is estimated by either maximizing a pseudolikelihood or solving an estimating equation. It is shown that this estimator is asymptotically normal, and a consistent estimator of its large sample variance is given. A simulation study shows that the proposed semiparametric method is better than the parametric ones available when the error distribution is unknown, which is almost always the case in practice. It turns out that there is no loss of asymptotic efficiency as a result of the estimation of regression parameters. An empirical example on portfolio management is used to illustrate the method.     

DELETE-2 AND DELETE-3 JACKKNIFE PROCEDURES FOR UNMASKING IN REGRESSION

Single-case deletion regression diagnostics have been used widely to discover unusual data points, but such approaches can fail in the presence of multiple unusual data points and as a result of masking. We propose a new approach to the use of single-case deletion diagnostics that involves applying these diagnostics to delete-2 and delete-3 jackknife replicates of the data, and considering the percentage of times among these replicates that points are flagged as unusual as an indicator of their influence. By considering replicates that exclude certain collections of points, subtle masking effects can be uncovered.     

NONLINEAR REGRESSION MODELLING AND TIME DEPENDENT COVARLATES IN REPEATED MEASURES EXPERIMENTS

The procedure of Verbyla & Cullis (1990) is extended to cater for the analysis of repeated measures data in which either non-linear modelling of the treatment contrasts is required and or there are time dependent covariates. These extensions are illustrated via two agricultural data sets.     

LOG-LINEAR MODELS FOR MEAN AND DISPERSION IN MIXED POISSON REGRESSION MODELS

This paper is concerned with the analysis of repeated measures count data overdispersed relative to a Poisson distribution, with the overdispersion possibly heterogeneous. To accommodate the overdispersion, the Poisson random variable is compounded with a gamma random variable, and both the mean of the Poisson and the variance of the gamma are modelled using log linear models. Maximum likelihood estimates (MLE) are then obtained. The paper also gives extended quasi-likelihood estimates for a more general class of compounding distributions which are shown to be approximations to the MLEs obtained for the gamma case. The theory is illustrated by modelling the determination of asbestos fibre intensity on membrane filters mounted on microscope slides.     

A NOTE ON MULTIVARIATE LOGISTIC MODELS FOR CONTINGENCY TABLES

The log-linear model is a tool widely accepted for modelling discrete data given in a contingency table. Although its parameters reflect the interaction structure in the joint distribution of all variables, it does not give information about structures appearing in the margins of the table. This is in contrast to multivariate logistic parameters, recently introduced by Glonek & McCullagh (1995), which have as parameters the highest order log odds ratios derived from the joint table and from each marginal table. Glonek & McCullagh give the link between the cell probabilities and the multivariate logistic parameters, in an algebraic fashion. The present paper focuses on this link, showing that it is derived by general parameter transformations in exponential families. In particular, the connection between the natural, the expectation and the mixed parameterization in exponential families (Barndorff-Nielsen, 1978) is used; this also yields the derivatives of the likelihood equation and shows properties of the Fisher matrix. The paper emphasises the analysis of independence hypotheses in margins of a contingency table.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.