A LINEAR MODEL WITH ERRORS LACKING A VARIANCE1

The problem discussed is that of estimating β= (β¹, …, β_k) in the model Y=βX +ε when X has a specified multivariate distribution and the error ε does not necessarily have a finite second moment, for example, ε symmetric stable. We construct a moment estimator based on the empirical characteristic function and establish asymptotic unbiassedness and normality. Most of the paper is concerned with the case when X is normal. Forms of the suggested estimator are given in (2.5), (4.6) and (5.5).     

RECURSIVE ESTIMATION OF THE GENERAL LINEAR MODEL WITH DEPENDENT ERRORS AND MULTIPLE ADDITIONAL OBSERVATIONS1

The techniques for recursive estimation of the general linear model with dependent errors and known second order properties, is generalised to allow for simultaneous addition of an arbitrary number of additional observations. Computational formulae for recursive updating of parameter estimates are derived, together with a sequence of univariate recursive residuals for testing the constancy of the regression relation over time.     

AN ASYMPTOTIC EXPANSION FOR THE VARIANCE OF THE ERRORS OF MISCLASSIFICATION OF THE LINEAR DISCRIMINANT FUNCTION1

A technique for deriving asymptotic expansions for the variances of the errors of misclassification of the linear discriminant function (Anderson's classification statistic) is developed. These expansions are shown to be in reasonable agreement with the sample values of the variances of the errors obtained from some sampling experiments.     

MAXIMUM LIKELIHOOD ESTIMATION IN LINEAR MODELS WITH EQUI-CORRELATED RANDOM ERRORS

Necessary and sufficient conditions for the existence of maximum likelihood estimators of unknown parameters in linear models with equi‐correlated random errors are presented. The basic technique we use is that these models are, first, orthogonally transformed into linear models with two variances, and then the maximum likelihood estimation problem is solved in the environment of transformed models. Our results generalize a result of Arnold, S. F. (1981) [The theory of linear models and multivariate analysis. Wiley, New York]. In addition, we give necessary and sufficient conditions for the existence of restricted maximum likelihood estimators of the parameters. The results of Birkes, D. & Wulff, S. (2003) [Existence of maximum likelihood estimates in normal variance‐components models. J Statist Plann. Inference. 113 , 35–47] are compared with our results and differences are pointed out.     

A NOTE ON THE MULTIVARIATE LINEAR MODEL WITH CONSTRAINTS ON THE DEPENDENT VECTOR

It is shown that the least squares estimators of B and Σ in the multivariate linear model {E Y _i= X ₁ B , D ( Y _i) =Σ, 1 ≤ i ≤ n , Y ₁ Y _n uncorrelated} subject to the constraints Y _i M = X _i N are just the usual least squares estimators = ( X'X )-¹ X'Y and ΣC = 1/n( Y-X )( Y-X ) in the unconstrained model where Σ has full rank. Tests of hypotheses concerning B are discussed for situations in which each Y _i has a multivariate normal distribution, and examples of the applicability of the model reviewed.     

INFLUENCE DIAGNOSTICS FOR THE NORMAL LINEAR MODEL WITH CENSORED DATA

Methods of detecting influential observations for the normal model for censored data are proposed. These methods include one-step deletion methods, deletion of observations and the empirical influence function. Emphasis is placed on assessing the impact that a single observation has on the estimation of coefficients of the model. Functions of the coefficients such as the median lifetime are also considered. Results are compared when applied to two sets of data.     

BAYESIAN ANALYSIS OF THE LINEAR REGRESSION MODEL WITH NON-NORMAL DISTURBANCES

This paper considers the Bayesian analysis of a linear regression model with identically independently distributed non-normal disturbances. The distribution of disturbances is approximated by an Edgeworth series distribution with cumulants, of order higher than fourth, negligible. The posterior distribution of the regression coefficients vector is obtained under the assumption of a g-prior distribution for the parameters of the model. The Bayes estimator and its Bayes risk of the estimator are derived under a quadratic loss structure.     

A NOTE ON ESTIMATING LINEAR TREND IN A REGRESSION MODEL WITH SERIALLY CORRELATED ERROR COMPONENTS

ABSTRACT

We consider a linear trend regression model when the disturbances follow a serially correlated one-way error component model. In this model, we investigate the performance of the Ordinary Least Squares Esitmator (OLSE), First Difference Estimator (FDE), Generalized Least Squares Estimator (GLSE) and the Cochrane-Orcutt-Transformation Estimator (COTE) of slope coefficient in terms of efficiency. The main findings are as follows: (1) when the autocorrelation is close to unity, then the FDE is approximately the GLSE; (2) the OLSE is better than the COTE; and (3) when the value of the autocorrelation is kept constant and T → ∞, the OLSE, COTE and GLSE are asymptotically equivalent whereas the FDE is worse than the other estimators in terms of efficiency.     

SPLINE SMOOTHER AS A DYNAMIC LINEAR MODEL

The paper shows how a finite dimensional representation of a cubic smoothing spline can be put in the framework of a dynamic linear model. The formulation provides an updating scheme when observations do not occur sequentially in time or space.     

NOTES ON CONFIDENCE REGIONS FOR A LINEAR MODEL

Several confidence regions for the means and standard deviation of a linear model are noted. Simulation to find the appropriate percentage points of sampling distributions is suggested as a practical way of constructing rectangular shaped regions.     

COMPONENTS OF VARIANCE MODELS WITH TRANSFORMATIONS

Power transformations are a popular way to improve the agreement between the observations and the assumptions in a statistical model. In this paper it is assumed that the data, after appropriate power transformation Λ, satisfies a variance components model, with independent Gaussian components. The focus is on inference for quantities which have an interpretation regardless of the choice of Λ (Carroll & Ruppert, 1981) – in particular the intraclass correlation coefficient ρ, the predicted probability of a new observation being less than a specified value and the predicted quantile. It is shown that, in the case Λ= 0, the asymptotic variance of ρ is the same, whether or not one treats Δ as estimated or as known. This supports an empirical conjecture of Solomon (1985). For predicted probabilities and predicted quantiles the variance when A is estimated is shown to be only slightly greater than the variance assuming Δ is known, except in the tails of the distribution where there can be substantial difference between the two variances.     

A-OPTIMAL RUN ORDERS WITH A LINEAR TREND

A-optimal ran orders are considered in the presence of a linear trend with emphasis on non-orthogonal situations where no trend-free run order can be. A-optimal. Some possibilities for further extension are also briefly indicated.     

VARIANCE ESTIMATION IN THE ERROR COMPONENTS REGRESSION MODEL

ABSTRACT

In a regression model with a random individual and a random time effect explicit representations of the nonnegative quadratic minimum biased estimators of the corresponding variances are deduced. These estimators always exist and are unique. Moreover, under normality assumption of the dependent variable unbiased estimators of the mean squared errors of the variance estimates are derived. Finally, confidence intervals on the variance components are considered.     

JACKKNIFING THE GENERAL LINEAR MODEL

The aim of this paper is to investigate the problems of estimating a smooth function of the parameters in a general linear model and to clarify some of the points raised by Hinkley (1977) in connection with this problem. An example of the type of problem at hand is that of estimating the maximum (or minimum) mean value in a quadratic regression model. The estimator based on the least squares estimator of the parameters in the linear model is compared to the jackknife estimator and the weighted jackknife estimator proposed by Hinkley (1977). The asymptotic properties of the estimators are examined and their small sample properties are compared through simulation studies.     

SAMPLING ERRORS OF GENETIC CORRELATION COEFFICIENTS CALCULATED FROM ANALYSES OF VARIANCE AND COVARIANCE

Summary. In this paper a formula is developed for estimating the sampling variance of a genetic correlation estimated from analyses of variance and covariance. The formula holds provided the heritability estimate of neither character is zero. However, the development assumes a constant number of offspring per sire, k , and the effect of varying values of k is discussed briefly. The efficiency of experiments from which genetic parameters are to be estimated has also been investigated and optimum values of k are given for various combinations of phenotypic and genetic parameters.     

ON SPLINE SMOOTHING WITH AUTOCORRELATED ERRORS

The generalised cross-validation criterion for choosing the degree of smoothing in spline regression is extended to accommodate an autocorrelated error sequence. It is demonstrated via simulation that the minimum generalised cross-validation smoothing spline is an inconsistent estimator in the presence of autocorrelated errors and that ignoring even moderate autocorrelation structure can seriously affect the performance of the cross-validated smoothing spline. The method of penalised maximum likelihood is used to develop an efficient algorithm for the case in which the autocorrelation decays exponentially. An application of the method to a published data-set is described. The method does not require the data to be equally spaced in time.     

CHANGE-OVER DESIGNS WITH ERRORS FOLLOWING A FIRST ORDER AUTOREGRESSIVE PROCESS

This paper deals with the problem of analysing the change over design in the context of a first order autoregressive process for the error terms. The method of maximum likelihood has been adopted for estimating treatment effects. The conditions derived for obtaining a balanced change over design show that a change over design balanced in the absence of autocorrelation is not necessarily balanced in the presence of autocorrelation. Also, it is observed that the autocorrelation co-efficient and the treatment effect when p≠0 can be tested as usual with the likelihood ratio test criterion.