Efficiency of iterated WLS in the linear model with completely unknown heteroskedasticity

Iterated weighted least squares (IWLS) is investigated for estimating the regression coefficients in a linear model with symmetrically distributed errors. The variances of the errors are not specified; it is not assumed that they are unknown functions of the explanatory variables nor that they are given in some parametric way.
IWLS is carried out in a random number of steps, of which the first one is OLS. In each step the error variance at time t is estimated with a weighted sum of m squared residuals in the neighbourhood of t and the coefficients are estimated using WLS. Furthermore an estimate of the co-variance matrix is obtained. If this estimate is minimal in some way the iteration process is stopped.
Asymptotic properties of IWLS are derived for increasing sample size n . Some particular cases show that the asymptotic efficiency can be increased by allowing more than two steps. Even asymptotic efficiency with respect to WLS with the true error variances can be obtained if m is not fixed but tends to infinity with n and if the heteroskedasticity is smooth.     

Analysis of variance on indirectly measured variables

Abstract  When observations from a normal distribution can only be obtained indirectly by counting the number of subjects responding to a previously chosen dose, parameter estimates can be obtained by using probit analysis. Well-known is the maximum likelihood technique of parameter estimation, less known is the approach by weighted least squares. The latter approach is followed to compare the parameters of several normal distributions by testing their equality, in analogy with the analysis of variance. A practical situation gave rise to this study and it is worked out at the end of the paper.     

Estimation in a growth study with irregular measurement times

Between 1982 and 1988 a growth study was carried out at the Division of Pediatric Oncology of the University Hospital of Groningen. A special feature of the project was that sample sizes are small and that ages at entry may be very different. In addition the intended design was not fully complied with. This paper highlights some aspects of the statistical analysis which is based on (1) reference scores, (2) statistical procedures allowing for an irregular pattern of measurement times caused by missing data and shifted measurement times.     

Combining Modelling Strategies to Analyse Teaching Styles Data

This paper combines two estimation procedures: Iterative Generalized Least Squares as used in the software MLwiN; Gibbs Sampling as employed in thesoftware BUGS to produce a modelling strategy that respects the hierarchical natureof the Teaching Styles data and also allows for the endogeneity problems encountered when examining pupil progress.     

Using Weighted Least Squares to Deflate Input–Output Tables

This article proposes a balancing procedure for the deflation of input–output (I-O) tables from the viewpoint of users. This is a ‘subjective’ variant of the Weighted Least Squares (WLS) method, already known in the literature. It is argued that it is more flexible than other methods, and it is shown that SWLS subsumes the first-order approximation of RAS as a special case. Flexibility is due to the facts that (a) users can attach differential ‘reliability’ weights to first (unbalanced) estimates, depending on the confidence they have in the different parts of their pre-balancing work, (b) differently from RAS, one is not bound to take any row or column total as exogenously given, and (c) additional constraints can be added to it. The article describes also how SWLS was utilised to estimate a yearly (1959–2000) series of constant-price I-O tables for the Italian economy.     

Cross Validation Calibration Based Shrinkage – Revisited for the Construction of Prognostic Indices

The use of shrinkage methods for the construction of prognostic indices has been paid increasing attention in the literature on medical statistics in the last years. One approach for the construction of a shrinkage factor is cross validation calibration as suggested by van H ouwelingen and le C essie (1990). We investigate this approach in more detail. First we try to clarify why shrinkage factors constructed by cross validation calibration tend to be smaller than 1. Second we explain why use of this shrinkage factor can result in an improvement of the average prediction error. Third we investigate the possible gain for constellations relevant in medical research by means of a simulation study, focusing on the dilemma, that the improvement on average has to be paid by distinct deteriorations for some patients. Finally we conclude that it is necessary to rethink the choice of loss functions in constructing prognostic indices before recommendations about the use of shrinkage methods can be made.     

Optimal versus orthogonal and equivalent-estimation design of blocked and split-plot experiments

This article provides an overview of the recent literature on the design of blocked and split-plot experiments with quantitative experimental variables. A detailed literature study introduces the ongoing debate between an optimal design approach to constructing blocked and split-plot designs and approaches where the equivalence of ordinary least squares and generalized least squares estimates are envisaged. Examples where the competing design strategies lead to totally different designs are given, as well as examples in which the optimal experimental designs are orthogonally blocked or equivalent-estimation split-plot designs.     

基于空间非均衡视角的大城市病成因研究

本文通过多次实验,选取空间自相关程度最为显著的距离阈值,设定空间权重矩阵,使得空间权重矩阵更加符合我国城市分布特征,充分体现人口在城市集聚过程中的空间自回归机制,并结合广义空间模型和广义空间两阶段最小二乘法,实证探查“大城市病”成因,研究发现:就业机会、流通经济布局和公共服务的空间非均衡是导致人口过度集聚,从而引致“大城市病”的显著相关因素,并针对以上因素给出了对策建议。     

Quasi-likelihood inference in a generalized linear mixed model for balanced data

For modelling the effect of crossed, fixed factors on the response variable in balanced designs with nested stratifications, a generalized linear mixed model is proposed. This model is based on a set of quasi-likelihood assumptions which imply quadratic variance functions. From these variance functions, deviances are obtained to quantify the variation per stratification. The effects of the fixed factors will be tested, an dispersion components will be estimated. The practical use of the model is illustrated by reanalysing a soldering failures problem.     

Least squares estimator for regression models with some deterministic time varying parameters

Here we study the least squares estimates in some regression models. We assume that the evolution of the parameter is linearly explosive (i.e. polynomial), or stable (i.e. sinusoidal). We prove the strong consistency, and establish the rate of convergence.     

On the backfitting algorithm for additive regression models

We analyse additive regression model fitting via the backfitting algorithm. We show that in the case of a large class of curve estimators, which includes regressograms, simple step-by-step formulae can be given for the back-fitting algorithm. The result of each cycle of the algorithm may be represented succinctly in terms of a sequence of d projections in n-dimensional space, where d is the number of design coordinates and n is sample size. It follows from our formulae that the limit of the algorithm is simply the projection of the data onto that vector space which is orthogonal to the space of all n-vectors fixed by each of the projections. The formulae also provide the convergence rate of the algorithm, the variance of the backfitting estimator, consistency of the estimator, and the relationship of the estimator to that obtained by directly minimizing mean squared distance.     

Contamination in linear regression models and its influence on estimators

The consequences of the omission of possibly contaminated observations in a linear regression model for the performance of the ordinary least squares ( LS- ) estimator are discussed. We compare the ordinary L Sestimator with the corresponding 'never pooled' LS -estimator with respect to the matrix-valued mean squared error. Necessary and sufficient conditions are derived for the superiority of an estimator to another one and tests are proposed to check these conditions. Finally the resulting preliminary-test-estimators are investigated.     

Inferential Causal Models: Integrating Quality & Quantity

In-depth data analysis plus statistical modeling can produce inferentialcausal models. Their creation thus combines aspects of analysis by close inspection,that is, reason analysis and cross-tabular analysis, with statistical analysis procedures,especially those that are special cases of the generalized linear model (McCullaghand Nelder, 1989; Agresti, 1996; Lindsey, 1997). This paper explores some of the roots of this combined method and suggests some new directions. An exercise clarifies some limitations of classic reason analysis by showing how the cross tabulation of variables with controls for test factors may produce better inferences. Then, given the cross tabulation of several variables, by explicating Coleman effect parameters, logistic regressions, and Poisson log-linear models, it shows how generalized linear models provide appropriate measures of effects and tests of statistical significance. Finally, to address a weakness of reason analysis, a case-control design is proposed and an example is developed.