EMPIRICAL LIKELIHOOD‐BASED INFERENCES FOR PARTIALLY LINEAR MODELS WITH MISSING COVARIATES

This paper considers statistical inference for partially linear models Y = X ^? β +ν(Z) +? when the linear covariate X is missing with missing probability π depending upon (Y, Z). We propose empirical likelihood‐based statistics to construct confidence regions for β and ν(z). The resulting empirical likelihood ratio statistics are shown to be asymptotically chi‐squared‐distributed. The finite‐sample performance of the proposed statistics is assessed by simulation experiments. The proposed methods are applied to a dataset from an AIDS clinical trial.     

A GEOMETRIC CHARACTERIZATION OF LINEAR REGRESSION

This paper concerns the asymptotic properties of the maximum likelihood estimators of the parameters in a non regular Cox model involving a change-point in the regression on time-dependent covariates. The global consistency derives from the uniform convergence of the partial log-likelihood. We prove that the estimator of the change-point is n -consistent and the estimator of the regression parameter n 1/2 -consistent, and their asymptotic distributions are established.     

THE BOOTSTRAP RISK OF LINEAR EMPIRICAL BAYES ESTIMATES

This paper discusses the bootstrap risk of the linear empirical Bayes estimate of the form θ=Ǎ+B̌x, where x is the current observation, and Ǎ and B̌ are generally functions of the estimates of the prior parameters. The standard error of this risk is developed and ‘computations’ of both the bootstrap risk and its standard error are made.     

SIZE AND SHAPE AS LINEAR COMBINATIONS OF UNTRANSFORMED MEASUREMENTS

A numerical specification of ‘size’ and ‘shape’ is of interest for making interpretations in morphometrics. Starting from a, possibly large, set m ₁,…, m_r of size measurements, e.g. m ₁= height, m ₂= sitting height, etc., a preliminary analysis provides the set x ₁,…,x_p of size measurements to be used, e.g. x ₁= m ₁? m ₂= subischial leg length, x ₂= m ₂= sitting height, and x ₃= head circumference. In general these x_j are constructed as appropriately scaled linear combinations of the original measurements. A constant term should not be included because size measurements have to be 0 if all x_j are 0. Our theory requires a (compromise) vector μof means and a matrix Σof (co)variances. Size being specified as an optimalsize characteristic of the form c ′ x , the remaining morphological information is expressed by, at most, p? 1 components of shapeof the form d ′ x. Relations with Darroch-Mosimann [9] Darroch, J. N. and Mosimann, J. E. 1985. Canonical and Principal Components of Shape. Biometrika, 72: 241–252. [Crossref], [Web of Science ®] , [Google Scholar]are indicated. An application to human growth is made and other applications are suggested.

Don't read my book, think for yourself.

C. R. Rao, personal communications, 1981     

MODEL‐BASED DIRECT ESTIMATION OF SMALL‐AREA DISTRIBUTIONS

Much of the small‐area estimation literature focuses on population totals and means. However, users of survey data are often interested in the finite‐population distribution of a survey variable and in the measures (e.g. medians, quartiles, percentiles) that characterize the shape of this distribution at the small‐area level. In this paper we propose a model‐based direct estimator (MBDE, Chandra and Chambers) of the small‐area distribution function. The MBDE is defined as a weighted sum of sample data from the area of interest, with weights derived from the calibrated spline‐based estimate of the finite‐population distribution function introduced by Harms and Duchesne, under an appropriately specified regression model with random area effects. We also discuss the mean squared error estimation of the MBDE. Monte Carlo simulations based on both simulated and real data sets show that the proposed MBDE and its associated mean squared error estimator perform well when compared with alternative estimators of the area‐specific finite‐population distribution function.     

SOME PROPERTIES OF A RANDOM LINEAR DIFFERENCE EQUATION1

Some simple conditions are given for the absolute continuity of the limiting distribution of a random linear difference equation. These results are applied to the super-critical Bellman-Harris branching process with immigration. When the coefficients of the difference equation are non-negative and there is no limiting distribution, it is shown that the asymptotic behaviour of the solutions is the same as that of the partial sums of a divergent random power series. A number of limit theorems are given for the latter situation.     

CRITICAL VALUE APPROXIMATIONS FOR TESTS OF LINEAR REGRESSION DISTURBANCES

Two important classes of tests for non-spherical disturbances in the linear regression model involve test statistics whose null distributions and hence critical values depend on the regressors. This paper investigates the accuracy of the normal, two moment beta and four moment beta approximations to the critical values of such tests. An empirical experiment aimed at evaluating the accuracy of the approximations for a variety of tests against autocorrelation and heteroscedasticity is conducted. Overall the approximations are found to provide reasonably accurate critical values with skewness being a factor determining the degree of accuracy.     

THE CONSTRUCTION OF EQUILEVERAGE DESIGNS FOR MULTIPLE LINEAR REGRESSION

The hat matrix is widely used as a diagnostic tool in linear regression because it contains the leverages which the independent variables exert on the fitted values. In some experiments, cases with high leverage may be avoided by judicious choice of design for the independent variables. A variety of methods for constructing equileverage designs for linear regression are discussed. Such designs remove one of the factors, namely large leverage points, which can lead to nonrobust estimators and tests. In addition, a method is given for combining equileverage designs to test for lack of fit of the linear model.     

FAST DOUBLE BOOTSTRAP TESTS OF NONNESTED LINEAR REGRESSION MODELS

ABSTRACT

It has been shown in previous work that bootstrapping the J test for nonnested linear regression models dramatically improves its finite-sample performance. We provide evidence that a more sophisticated bootstrap procedure, which we call the fast double bootstrap, produces a very substantial further improvement in cases where the ordinary bootstrap does not work as well as it might. This FDB procedure is only about twice as expensive as the usual single bootstrap.     

FAST DOUBLE BOOTSTRAP TESTS OF NONNESTED LINEAR REGRESSION MODELS

It has been shown in previous work that bootstrapping the J test for nonnested linear regression models dramatically improves its finite-sample performance. We provide evidence that a more sophisticated bootstrap procedure, which we call the fast double bootstrap, produces a very substantial further improvement in cases where the ordinary bootstrap does not work as well as it might. This FDB procedure is only about twice as expensive as the usual single bootstrap.     

DETECTION AND TESTING OF DIFFEFENT TYPES OF OUTLIER IN LINEAR STRAUCTURAL RELATIONASHIPS

The linear structural model provides one way of modelling a linear relationship between two random variables. It is well known that problems of unidentifiability arise for unreplicated observations and normal error structure. As in all data sets, outliers can arise and methods are needed for detecting and testing them. An outlier-generating model of mean–slippage type can be used to characterise four different forms of outlier manifestation. It is interesting to find that the unidentifiability problem provides no obstacle for detecting or testing the outliers for three of the four forms. Detection principles, and specific discordancy tests, are derived and illustrated by application to some data on physical measurements of Pacific squid.     

MINIMUM MSE ESTIMATION BY LINEAR COMBINATIONS OF ORDER STATISTICS1

In this paper we assume that in a random sample of size ndrawn from a population having the pdf f(x; θ) the smallest r₁ observations and the largest r₂ observations are censored (r₁0, r₂0). We consider the problem of estimating θ on the basis of the middle n-r₁-r₂ observations when either f(x;θ)=θ^-1f(x/θ) or f(x;θ) = (aθ)¹f(x-θ)/aθ) where f(·) is a known pdf, a (<0) is known and θ (>0) is unknown. The minimum mean square error (MSE) linear estimator of θ proposed in this paper is a “shrinkage” of the minimum variance linear unbiased estimator of θ. We obtain explicit expressions of these estimators and their mean square errors when (i) f(·) is the uniform pdf defined on an interval of length one and (ii) f(·) is the standard exponential pdf, i.e., f(x) = exp(–x), x0. Various special cases of censoring from the left (right) and no censoring are considered.     

PERFORMANCE OF THE LOCATION LINEAR DISCRIMINANT FUNCTION UNDER ACROSS-LOCATION HETEROSCEDASTICITY

ABSTRACT

Classification of data consisting of both categorical and continuous variables between two groups is often handled by the sample location linear discriminant function confined to each of the locations specified by the observed values of the categorical variables. Homoscedasticity of across-location conditional dispersion matrices of the continuous variables is often assumed. Quite often, interactions between continuous and categorical variables cause across-location heteroscedasticity. In this article, we examine the effect of heterogeneous across-location conditional dispersion matrices on the overall expected and actual error rates associated with the sample location linear discriminant function. Performance of the sample location linear discriminant function is evaluated against the results for the restrictive classifier adjusted for across-location heteroscedasticity. Conclusions based on a Monte Carlo study are reported.     

LINEAR MINIMAX ESTIMATORS FOR ESTIMATING A FUNCTION OF THE PARAMETER

In this note we provide sufficient conditions for the minimaxity of linear estimators of the form aX+b in the one-parameter exponential family for estimating a differentiable function g(θ) with normalized quadratic loss. We provide some examples which show that the natural estimator X is minimax in estimating a function of the parameter (different from the mean).     

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.     

ESTIMATION OF LLUCH'S EXTENDED LINEAR EXPENDITURE SYSTEM FROM CROSS-SECTIONAL DATA1

Complete sets of demand relations may be fitted using varying types of sample information and varying a priori specifications. In this paper the identification and estimation of Lluch's extended linear expenditure system (ELES) from cross-sectional data alone is investigated. Under the most favourable conditions of data availability, all of the parameters of the ELES model are identified, and are estimable by the method of reduced form least squares. This is the case where observations on permanent income are available for the consuming units of the cross section and where, in addition, prices are recorded (even though they do not vary from one consuming unit to the next). Under the least favourable conditions only the marginal budget shares are identified. This corresponds to the case where no data on permanent income, or on savings, are available. The conventional ordinary least squares estimators of the marginal budget shares are, under these conditions, biased and inconsistent. Expressions are developed for the large-sample biases.     

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.