Abstract: Efficient method of moments estimation techniques include many commonly used techniques, including ordinary least squares, two- and three-stage least squares, quasi maximum likelihood, and versions of these for nonlinear environments. For models estimated by any efficient method of moments technique, the authors define analogues to the maximum likelihood based Wald, likelihood ratio, Lagrange multiplier, and minimum chi-squared statistics. They prove the mutual asymptotic equivalence of the four in an environment that allows for disturbances that are auto correlated and heteroskedastic. They also describe a very convenient way to test a linear hypothesis in a linear model. Copyright 1987 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Abstract: A class of robust estimates for the linear model is introduced. These estimates, called MM-estimates, have simultaneously the following properties: (i) they are highly efficient when the errors have a normal distribution and (ii) their breakdown-point is 0.5. The MM-estimates are defined by a three-stage procedure. In the first stage an initial regression estimate is computed which is consistent robust and with high breakdown-point but not necessarily efficient. In the second stage an M-estimate of the errors scale is computed using residuals based on the initial estimate. Finally, in the third stage an M-estimate of the regression parameters based on a proper redescending psi-function is computed. Consistency and asymptotical normality of the MM-estimates assuming random carriers are proved. A convergent iterative numerical algorithm is given. Finally, the asymptotic biases under contamination of optimal bounded influence estimates and MM-estimates are compared.

Abstract: This paper considers estimation and hypothesis tests for coefficients of linear regression models, where the coefficient estimates are based on location measures defined by an asymmetric least squares criterion function. These asymmetric least squares estimators have properties which are analogous to regression quantile estimators, but are much simpler to calculate, as are the corresponding test statistics. The coefficient estimators can be used to construct test statistics for homoskedasticity and conditional symmetry of the error distribution, and we find these tests compare quite favorably with other commonly-used tests of these null hypotheses in terms of local relative efficiency. Consequently, asymmetric least squares estimation provides a convenient and relatively efficient method of summarizing the conditional distribution of a dependent variable given the regressors, and a means of testing whether a linear model is an adequate characterization of the "typical value" for this conditional distribution.

Abstract: Generalized additive models have the form η(x) = α + σ fj (x j ), where η might be the regression function in a multiple regression or the logistic transformation of the posterior probability Pr(y = 1 | x) in a logistic regression. In fact, these models generalize the whole family of generalized linear models η(x) = β′x, where η(x) = g(μ(x)) is some transformation of the regression function. We use the local scoring algorithm to estimate the functions fj (xj ) nonparametrically, using a scatterplot smoother as a building block. We demonstrate the models in two different analyses: a nonparametric analysis of covariance and a logistic regression. The procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis. A variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions: these include confidence curves, degrees of freedom estimates, and approximat...

Abstract: The commonly used linear K-l and K-e models of turbulence are shown to be incapable of accurately predicting turbulent flows where the normal Reynolds stresses play an important role. By means of an asymptotic expansion, nonlinear K-l and K-e models are obtained which, unlike all such previous nonlinear models, satisfy both realizability and the necessary invariance requirements. Calculations are presented which demonstrate that this nonlinear model is able to predict the normal Reynolds stresses in turbulent channel flow much more accurately than the linear model. Furthermore, the nonlinear model is shown to be capable of predicting turbulent secondary flows in non-circular ducts - a phenomenon which the linear models are fundamentally unable to describe. An additional application of this model to the improved prediction of separated flows is discussed briefly along with other possible avenues of future research.

Abstract: 1. An Up-Dated Viewpoint: Cell Means Models. 2. Basic Results for Cell Means Models: The 1-Way Classification. 3. Nested Classifications. 4. The 2-Way Crossed Classification with All-Cells-Filled Data: Cell Means Models. 5. The 2-Way Classification with Some-Cell Empty Data: Cell Means Models. 6. Models with Covariables (Analysis of Covariance): the 1-Way Classification. 7. Matrix Algebra and Quadratic Forms ( A Prelude to Chapter 8). 8. A General Linear Model. 9. The 2-Way Crossed Classification: Overparameterized Models. 10. Extended Cell Means Models. 11. Models with Covariables: The General Case and Some Applications. 12. Comments on Computing Packages. 13. Mixed Models: A Thumbnail Survey. References. Statistical Tables. List of Tables and Figures. Index.

Abstract: Andersen (1970) considered the problem of inference on random effects linear models from binary response panel data. He showed that inference is possible if the disturbances for each panel member are known to be white noise with the logistic distribution and if the observed explanatory variables vary over time. A conditional maximum likelihood estimator consistently estimates the model parameters up to scale. The present paper shows that inference remains possible if the disturbances for each panel member are known only to be time-stationary with unbounded support and if the explanatory variables vary enough over time. A conditional version of the maximum score estimator (Manski, 1975, 1985) consistently estimates the model parameters up to scale.

Abstract: : This paper studies estimation of the parameters of generalized linear models in canonical form when the explanatory vector is measured with independent normal error. For the functional case, i.e., when the explanatory vectors are fixed constants, unbiased score functions are obtained by conditioning on certain sufficient statistics. This work generalizes results obtained for logistic regression. In the case that the explanatory vectors are independent and identically distributed with unknown distribtuion, efficient score functions are obtained using the theory developed in Begun et al. (1983). Keywords: Conditional score function; Efficient score function; Functional model; Generalized linear model; Measurement error; Structural model.

Abstract: SUMMARY This paper exploits the one step approximation, derived by Pregibon (1981), for the changes in the deviance of a generalized linear model when a single case is deleted from the data This approximation suggests a particular set of residuals which can be used, not only to identify outliers and examine distributional assumptions, but also to calculate measures of the influence of single cases on various inferences that can be drawn from the fitted model using likelihood ratio statistics Regression diagnostics for the Normal linear model are now well established in the literature They are comprehensively surveyed by Cook and Weisberg (1982) Many of these diagnostics use statistics which measure the effects of deleting single cases from the data These statistics exploit the exact algebraic relationship between the least squares fit of the linear model to a complete set of n cases, and the fit to the n - 1 cases remaining after the deletion of a single case The maximum likelihood (ML) estimation of most generalized linear models (GLMs) requires iterative methods The ML estimates from n - 1 cases cannot then be obtained as explicit functions of the results of the fit to all n cases In an important paper Pregibon (1981) derives useful one step approximations for the changes in the ML estimate and the deviance of the model when a single case is deleted, and he discusses some diagnostic methods which use these approximations Cook and Weisberg (1982) discuss GLM diagnostics briefly in Section 54 and they make use of Pregibon's results McCullagh and Nelder (1983) discuss diagnostics in their chapter on model checking (chapter 11) but Pregibon's approximations are not mentioned In this paper I describe some GLM diagnostics which all make use of Pregibon's one step approximations for the change in the components of the deviance when a single case is deleted Section 2 establishes the notation of the paper, Section 3 states the one step approximations, and Section 4 shows how these approximations can be used to define residuals and to measure the influence of individual cases on different aspects of the fitted model Throughout these three sections the computations will be exemplified using quantal assay data taken from Table V of Irwin (1937) These data were chosen for two reasons Firstly they have been reanalysed by Copenhaver and Mielke (1977) and Morgan (1985) and they are known to contain some features of interest Secondly, as they comprise only five cases, a variety of single case diagnostics can be completely tabulated and compared without taking up much space These data are

Abstract: Linear combinations of order statistics, or L-estimators, have played an extremely important role in the development of robust methods for the one-sample problem. We suggest analogs of L-estimators for the parameters of the linear model based on the p-dimensional “regression quantiles” proposed by Koenker and Bassett (1978). A uniform, Bahadur-type asymptotic representation of regression quantiles is established, and this yields a general asymptotic theory of L-estimators for the linear model. A leading example of the proposed estimators is an analog of the trimmed mean (TRQ), which is asymptotically equivalent to the trimmed least squares estimator studied by Ruppert and Carroll (1980), but appears to be somewhat less sensitive to influential design observations. This estimator is also asymptotically equivalent to the well-known Huber M-estimator, but offers the significant advantage that it is scale invariant. We illustrate the methods by reconsidering a mid-18th century linear model analyzed b...

Abstract: New models for multiple time series are introduced and illustrated in an application to international currency exchange rate data. The models, based on matrix-variate normal extensions of the dynamic linear model (DLM), provide a tractable, sequential procedure for estimation of unknown covariance structure between series. A principal components analysis is carried out providing a basis for easy model assessment. A practically important elaboration of the model incorporates time- variation in covariance matrices.

Abstract: Methods of identifying a series of piecewise linear models which approximate to a non-linear system over some operating range are investigated. Spatial linear models and models with signal-dependent parameters are considered.

Abstract: In a general univariate linear model, $M$-estimation of a subset of parameters is considered when the complementary subset is plausibly redundant. Along with the classical versions, both the preliminary test and shrinkage versions of the usual $M$-estimators are considered and, in the light of their asymptotic distributional risks, the relative asymptotic risk-efficiency results are studied in detail. Though the shrinkage $M$-estimators may dominate their classical versions, they do not, in general, dominate the preliminary test versions.

Abstract: Estimates of the linear model parameters in a linear transformation model with unknown increasing transformation are obtained by maximizing a partial likelihood. A resampling scheme (likelihood sampler) is used to compute the maximum partial likelihood estimates. It is shown that for a certain "local" parameter set where the "signal to noise ratio" is small, it is asymptotically possible to estimate the linear model parameters using the partial likelihood as well as if the transformation were known. In the case of the power transformation model with symmetric error distribution, this result is shown to also hold when the distribution of the error in the transformed linear model is unknown and is estimated. Monte Carlo results are used to show that for moderate sample size and small to moderate signal to noise ratio, the asymptotic results are approximately in effect and thus the partial likelihood estimates perform very well. Estimates of the transformation are introduced and it is shown that the estimates, when centered at the transformation and multiplied by $\sqrt{n}$, converge weakly to Gaussian processes.

Abstract: Until now, the problem of fitting self-stimulation rate-frequency functions has been dealt with by using linear models applied to the linear portion of the empirical curve. In this article, an alternative procedure is presented, together with three sigmoid growth models that seem to accurately fit rate-frequency data. From any of these models, it is possible to compute the two indices of stimulation efficacy in use in the parametric study of brain stimulation reward (M50 and theta 0), in addition to the inflection point of the curve, which can be used as an alternative to M50. Important relations allowing initial estimation of each parameter are provided, allowing use of computer programs derived from the Gauss-Newton algorithm for nonlinear regression. The considerations relevant to the choice of a nonlinear model are discussed in terms of each efficacy index.

Abstract: The category of multilinear models is analyzed that embraces several approaches developed in apparently independent references. Multilinear modeling is a means of describing a nonlinear system with a set of linear models. This is performed by distributing the inflow to the system (river reach) into inflows to submodels and treating these inflows with a family of different linear operators. The models considered are nearly as simple as linear models and almost as accurate as nonlinear hydrodynamical models.

Abstract: This paper examines the combined effects of multicollinearity, parameter stability, and alternative function forms in hedonic regression models. The results indicate that the significance and stability of the regression coefficients as well as prediction accuracy are sensitive to the choice of functional form and estimation technique. In certain respects nonlinear models proved to be more effective than linear models and ridge regression techniques were generally superior to OLS estimation. Since no single estimation technique or functional form was superior in all respects, the appraiser may have to choose between minimizing the average prediction error or maximizing prediction stability.

Abstract: We derive the information matrix test, suggested by White, for the normal fixed regressor linear model, and show that the statistic decomposes asymptotically into the sum of three independent quadratic forms. One of these is White's general test for heteroscedasticity and the remaining two components are quadratic forms in the third and fourth powers of the residuals respectively. Our results show that the test will fail to detect serial correlation and never be asymptotically optimal against heteroscedasticity, skewness and non-normal kurtosis.

Abstract: The least median of squared residuals regression line (or LMS line) is that line y = ax + b for which the median of the residuals |yi - axi - b |2 is minimized over all choices of a and b. If we rephrase the traditional ordinary least squares (OLS) problem as finding the a and b that minimize the mean of | yi - axi - b |2, one can see that in a formal sense LMS just replaces a “mean” by a “median.” This way of describing LMS regression does not do justice to the remarkable properties of LMS. In fact, LMS regression behaves in ways that distinguish it greatly from OLS as well as from many other methods for robustifying OLS (see, e.g., Rousseeuw 1984). As illustrations given here show, the LMS regression line should provide a valuable tool for studying those data sets in which the usual linear model assumptions are violated by the presence of some (not too small) groups of data values that behave distinctly from the bulk of the data. This feature of LMS regression is illustrated by the fit given in...

Abstract: The article considers the efficiency of ordinary least squares (OLS) coefficient estimates relative to generalized least squares (GLS) in a linear regression model where the disturbances follow a spatial autocorrelation pattern. The main result is that the limiting relative efficiency of OLS, as correlation among disturbances increases, tends to either zero or unity for most design matrices and correlation patterns encountered in practice. Which of these cases applies depends on the eigenvector corresponding to the largest eigenvalue of the spatial dependence matrix from the spatial autocorrelation scheme. In particular, the limiting relative efficiency of OLS is unity if this eigenvector lies in the column space of the design matrix. In practice, the relevant eigenvector will often be a column of ones. This implies that OLS is in the limit as good as GLS whenever there is a constant in the regression. We conclude, however, from several concrete examples that the loss in efficiency can still be s...

Abstract: Analytical modeling of structures subjected to ground motions is an important aspect of fully dynamic earthquake-resistant design. In general, linear models are only sufficient to represent structural responses resulting from earthquake motions of small amplitudes. However, the response of structures during strong ground motions is highly nonlinear and hysteretic. System identification 1s an effective tool for developing analytical models from experimental data. Testing of full-scale prototype structures remains the most realistic and reliable source of inelastic seismic response data. Pseudo-dynamic testing is a recently developed quasi-static procedure for subjecting full-scale structures to simulated earthquake response. The present study deals with structural modeling and the determination of optimal linear and nonlinear models by applying system identification techniques to elastic and inelastic pseudo-dynamic data from a full-scale, six-story steel structure. It is shown that the feedback of experimental errors during the pseudo-dynamic tests significantly affected the higher modes and led to an effective negative damping for the third mode. The contributions of these errors are accounted for and the small-amplitude modal properties of the test structure are determined. These properties are in agreement with the values obtained from a shaking table test of a 0.3 scale model. The nonlinear hysteretic behavior of the structure during strong ground motions is represented by a general class of Masing models. A simple model belonging to this class is chosen. with parameters which can be estimated theoretically, thereby making this type of model potentially useful during the design stages. The above model is identified from the experimental data and then its prediction capability and application in seismic design and analysis are examined.

Abstract: A common problem in the use of singly-constrained spatial interaction shopping models has been that of finding optimal parameter values. This problem has been exacerbated where improvements to the model have involved extra parameters to be estimated. In this paper it is shown that calibration of quite complex models can be achieved through modification of the conventional ‘gravity’ model to a generalised linear model with Poisson error structure and logarithmic link function. Data on observed trips between fifteen residential zones and eighty-three shopping destinations in Cardiff are used to test several models through application of the GLIM computing package. Models involving extra explanatory variables, origin-specific distance-decay parameters, and competing-destinations terms are all shown to offer worthwhile improvements in performance over the conventional singly-constrained model. An individual-specific model is also tested for a small sample of shoppers. Finally, some comments are made concernin...

Abstract: Consider the standard linear model where x x ,x 2 … are assumed to be the known p-vectors, β the unknown p-vector of regression coefficients, and e 1 , e 2 , …the independent random error sequence, each having a median zero Define the minimum L 1 norm estimator as,the solution of the minimization problem inf It is proved in this paper that is asymptotically normal under very weak conditions In particular, the condition imposed on {xi} is exactly the same which ensures the asymptotic normality of least-squares estimate:

Abstract: In regression analysis, the response is often transformed to remove heteroscedasticity and/or skewness. When a model already exists for the untransformed response, then it can be preserved by applying the same transform to both the model and the response. This methodology, which we call “transform both sides,” has been applied in several recent papers and appears highly useful in practice. When a parametric transformation family such as the power transformations is used, then the transformation can be estimated by maximum likelihood. The maximum likelihood estimator, however, is very sensitive to outliers. In this article, we propose diagnostics to indicate cases influential for the transformation or regression parameters. We also propose a robust bounded-influence estimator similar to the Krasker-Welsch regression estimator:

Abstract: SUMMARY Repeated-measures experiments involve two or more intended measurements per subject. If the within-subjects design is the same for each subject and no data are missing, then the analysis is relatively simple and there are readily available programs that do the analysis automatically. However, if the data are incomplete, and do not have the same arrangement for each subject, then the analysis becomes much more difficult. Beginning with procedures that are not optimal but are comparatively simple, we discuss unbalanced linear model analysis and then normal maximum likelihood (ML) procedures. Included are ML and REML (restricted maximum likelihood) estimators for the mixed model and also estimators for a model that allows arbitrary within-subject covariance matrices. The objective is to give procedures that can be implemented with available software.