Showing papers on "Linear model published in 1981"
TL;DR: This book brings together a number of procedures developed for regression problems in current use and includes material that either has not previously appeared in a textbook or if it has appeared is not generally available.
Abstract: This book brings together a number of procedures developed for regression problems in current use. Since the emphasis is on practical application theoretical results are stated without proofs in many cases. This book provides a standard basic course in multiple linear regression but it also includes material that either has not previously appeared in a textbook or if it has appeared is not generally available. Chapters 1 and 3 together provide a course in fitting a straight line without using matrix algebra at all. If chapter 2 is added the idea of matrix representation of regression problems can be introduced as well. Chapter 4 covers 2 predictor variables and chapter 5 deals with more complicated models. Selecting the best regression equation is discussed in chapter 6. Chapter 7 covers specific problems. Chapters 8 and 9 discuss 1) multiple regression and mathematical model building and 2) multiple regression applied to analysis of variance problems. Chapter 10 contains an introduction to nonlinear estimation. The 2nd edition contains many new regression ideas and techniques. In particular new computational algorithms and new software regression packages have made it very easy to investigate the allequacy of conjectured models with many different techniques.
3,676 citations
TL;DR: In this article, the authors considered consistency properties of the Box-Cox estimates (MLE's) of λ and the parameters in the linear model, as well as the asymptotic variances of these estimates.
Abstract: Following Box and Cox (1964), we assume that a transform Z i = h(Yi , λ) of our original data {Yi } satisfies a linear model. Consistency properties of the Box-Cox estimates (MLE's) of λ and the parameters in the linear model, as well as the asymptotic variances of these estimates, are considered. We find that in some structured models such as transformed linear regression with small to moderate error variances, the asymptotic variances of the estimates of the parameters in the linear model are much larger when the transformation parameter λ is unknown than when it is known. In some unstructured models such as transformed one-way analysis of variance with moderate to large error variances, the cost of not knowing λ is moderate to small. The case where the error distribution in the linear model is not normal but actually unknown is considered, and robust methods in the presence of transformations are introduced for this case. Asymptotics and simulation results for the transformed additive two-way ...
561 citations
Book•
1 Jan 1981TL;DR: In this paper, the regression quantile statistics of Koenker and Bassett (1978) are employed to construct an estimate of the error quantile function in linear models with iid errors.
Abstract: The regression quantile statistics of Koenker and Bassett (1978) are employed to construct an estimate of the error quantile function in linear models with iid errors. Some finite sample properties and the asymptotic behavior of the proposed estimator are derived. Comparisons with procedures based on residuals are made. The stackloss data of Brownlee (1965) is reanalyzed to illustrate the technique
258 citations
1 Oct 1981
TL;DR: In this paper, the problem of determining airplane model structure is addressed using linear and stepwise regressions, and the MSR was constructed to force a linear model for the aerodynamic coefficient first, then add significant nonlinear terms and delete nonsignificant terms from the model.
Abstract: The linear and stepwise regressions are briefly introduced, then the problem of determining airplane model structure is addressed. The MSR was constructed to force a linear model for the aerodynamic coefficient first, then add significant nonlinear terms and delete nonsignificant terms from the model. In addition to the statistical criteria in the stepwise regression, the prediction sum of squares (PRESS) criterion and the analysis of residuals were examined for the selection of an adequate model. The procedure is used in examples with simulated and real flight data. It is shown that the MSR performs better than the ordinary stepwise regression and that the technique can also be applied to the large amplitude maneuvers.
126 citations
Book•
1 Jan 1981TL;DR: A survey of recent developments in robust estimation and inference is directed primarily toward econometricians as mentioned in this paper, where it is argued that many of the techniques in common use in econometrics are highly sensitive to unverified hypotheses.
Abstract: This survey of recent developments in robust estimation and inference is directed primarily toward econometricians. It is argued that many of the techniques in common use in econometrics are highly sensitive to unverified hypotheses. Recent progress in designing alternative robust procedures is described and some prospects for future developments are discussed.
106 citations
101 citations
TL;DR: In this article, the Mihailov stability criterion is used to improve the Pade approximation method for linear model reduction, and the stability of the reduced model is assured if the original system is stable.
Abstract: This paper uses the property of the Mihailov stability criterion to improve the Pade approximation method for linear model reduction. Therefore the stability of the reduced model is assured, if the original system is stable. This method provides several different reduced models depending upon the constant k 2 to be chosen. It is rather simple, computationally very straightforward, and can be used for multi-input multi-output systems and unstable systems. Finally this paper introduces a method for estimating the order of the reduced model, and gives a possibility for solving the model reduction problem over a desired low-frequency interval. Numerical examples and comparison among different reduced models are given.
97 citations
1 Sep 1981
TL;DR: In this article, a brief introduction to the more common geometric structures is given, also showing their linear counterparts, and how these structures arise in systems theory by introducing the nonlinear control problems involved with mechanical manipulators, electrical networks, rotating electrical machinery and attitude control of spacecraft.
Abstract: The predominance of linear models in systems theory has tended to obscure the natural structure possessed by given nonlinear physical systems, either through linearisation, model order reduction, or choice of co-ordinates. The purpose of the paper is to motivate the reintroduction of geometric structure into systems theory. First, a brief introduction to the more common geometric structures is given, also showing their linear counterparts. It is then shown how these structures arise in systems theory by introducing the nonlinear control problems involved with mechanical manipulators, electrical networks, rotating electrical machinery and attitude control of spacecraft. The paper is concluded by considering the application of some of the geometric structures to nonlinear Hamiltonian and potential input/output systems.
80 citations
TL;DR: In this paper, the analysis of transformation of observations in the linear model with normal errors proposed by Box & Cox (1964) is considered, and a different choice of noninformative unnormed prior is advocated, which is not outcome dependent.
Abstract: SUMMARY The analysis of transformation of observations in the linear model with normal errors proposed by Box & Cox (1964) is considered. A different choice of noninformative unnormed prior is advocated, which is not outcome dependent. This new selection of prior leads to a formal identity between likelihood and Bayesian inference, both for the estimation of the best transformation to normality and for the presence of homoscedasticity and additivity under this transformation. Extension to a related problem is mentioned.
[...]
TL;DR: In this paper, the authors describe the observed values for each individual by a linear model in which the parameter vector of each individual is independently drawn from a multivariate normal distribution and the variance of the normalized linear combination of expected relative deviations from the population mean is obtained.
Abstract: A population is said to track with respect to a particular observable characteristic if, for each individual, the expected value of the relative deviation from the population mean remains unchanged over time. We describe the observed values for each individual by a linear model in which the parameter vector for each individual is independently drawn from a multivariate normal distribution. The variance of the normalized linear combination of expected relative deviations from the population mean is obtained. The proportion of the total variance (apart from random within-subject error) accounted for by this linear combination, less the variance of this linear combination in the case of a diagonal covariance matrix, is defined as the 'index of tracking'. When standardized variables are used, this index is the average of the pairwise correlation coefficients (corrected for within-subject error). Estimators are given, and jackknifing is used to obtain sampling distributions.
TL;DR: Recursive estimation techniques for fixed and completely random models are extended to mixed linear models by using the Kalman filter to obtain recursive estimators for a two-part random model where the second random factor obeys a generalized autoregressive process.
Abstract: Recursive estimation techniques for fixed and completely random models are extended to mixed linear models The Kalman filter is used to obtain recursive estimators for a two-part random model where the second random factor obeys a generalized autoregressive process By passing to the limit in an appropriate way, recursions for the mixed model are derived
TL;DR: In this article, the authors compare the relative merits of AR and window spectral estimation, the basis of the work being an extensive simulation of series constructed from a variety of models, and provide valuable information on the relative performance of various order determination criteria when applied to different types of models and varying data lengths.
Abstract: SUMMARY a process with a "mixed" spectrum. The paper also includes some discussion of two different methods of estimating the coefficients of AR models (the Burg method and the Yule-Walker approach), and of the performance of various order determination criteria, such as FPE, AIC and CAT. 1. I NTRODUCTION The problem of fitting finite parameter linear models to time series has recently aroused renewed interest, particularly in relation to the use of "automatic" model order determination procedures such as Akaike's AIC and Parzen's CAT criteria. Such models can be used to provide "parametric" estimates of the spectral density function, and, in particular, the method of "autoregressive spectral estimation" has attracted growing interest as an alternative to the more traditional non-parametric "window methods". In this paper we compare the relative merits of AR and "window" spectral estimation, the basis of the work being an extensive simulation of series constructed from a variety of models. The simulation study also provides valuable information on the relative performance of various order determination criteria when applied to different types of models and varying data lengths. Our conclusions are summarized in Section 7.
1 Dec 1981
TL;DR: The main lines of research undertaken during the period are: Probability Theory: Major advances were made in obtaining Edgeworth expansions in a variety of situations, e.g., involving discrete variables, and errors in variables models as mentioned in this paper.
Abstract: : The main lines of research undertaken during the period are: Probability Theory: Major advances were made in obtaining Edgeworth expansions in a variety of situations, e.g., involving discrete variables, and errors in variables models. New limit theorems were established and their applications were discussed. Several contributions have been made to characterization theory. Linear Models and Time Series: New methods of forecasting were developed using dynamic linear models and multiple bilinear time series models. Multivariate Analysis: Topics of research in this area included inference on interclass and intraclass correlations and principal component analysis. M-estimation: A unified theory of robust inference (estimation and tests of hypotheses) was developed using a convex discrepancy function for minimization.
TL;DR: In this article, Thiele formulated the canonical form of the linear model with normally distributed errors and reduced the general linear model to canonical form by means of an orthogonal transformation, thereby providing a new justification for the method of least squares.
Abstract: Summary The background for the work of T.N. Thiele (1838-1910) is sketched. He made contributions to the theory of skew distributions, he defined the cumulants and investigated their properties, and he formulated the canonical form of the linear model with normally distributed errors and reduced the general linear model to canonical form by means of an orthogonal transformation thereby providing a new justification for the method of least squares. Furthermore, he formulated the basic principle for the one-way analysis of variance and tested the significance of the variation between groups by means of the difference of the between-group variance and the within-group variance. He derived estimates of the parameters in the two-way classification model without interaction, also for the case with missing observations. He also extended the linear model with normally distributed errors by adding a 'quasisystematic' component, i.e. a Brownian motion, and he solved the estimation and prediction problems in this time series model, theoretically and numerically, by the method of least squares. Finally he stressed the importance of criticism of the model by means of graphical and numerical analysis of residuals using rational subgroups of observations.
TL;DR: This work presents a relatively simple alternative method for assessing the accuracy of the first-order Bonferroni upper bound that can be applied to any linear model and is suitable for routine use.
Abstract: At present, the first-order Bonferroni upper bound is the only practically useful tool for determining approximate critical values or p values for the maximum absolute studentized residual as a criterion for detecting a single outlier in a linear model. Available methods for assessing the accuracy of this bound require numerical integration and are difficult to apply routinely. We present a relatively simple alternative method that can be applied to any linear model and is suitable for routine use. The application to analyses of 2 m factorial experiments and regression models is illustrated with several examples.
TL;DR: In this paper, a large sample method for estimating the slope parameter in a linear model by minimizing a loss function related to the empirical cumulant generating function of the error distribution is presented.
Abstract: SUMMARY The paper presents a large sample method for estimating the slope parameter in a linear model by minimizing a loss function related to the empirical cumulant generating function of the error distribution. A family of estimators, indexed by a real parameter, is obtained and consistency and asymptotic normality established. The optimum member of the family is that which has minimum variance with respect to the parameter. This minimization together with a characterization result for the normal distribution leads to a procedure for the identification of outliers with respect to least squares.
Book•
1 Nov 1981
TL;DR: This volume deals with the increasingly complicated forms of linear models: single-equity models, multi-equation models, and models with unobserved variables and measurement error.
Abstract: Linear models attempt to state causal laws, thought to be operative in one or more groups, organizations, or nations. These models represent the researcher's idea about the structure according to which explanatory or independent variables combine to produce variations in response (or dependent) variables. This volume deals with the increasingly complicated forms of linear models: single-equation models, multi-equation models, and models with unobserved variables and measurement error.
TL;DR: The aim of this paper is to explain and to stress the importance of mathematical models and, in particular, linear models in the field of psychiatric epidemiology.
Abstract: The aim of this paper is to explain and to stress the importance of mathematical models and, in particular, linear models in the field of psychiatric epidemiology. An awareness of linear models is essential in contemplating the use of statistics of analyse epidemiological survey data, and in many cases the appropriate linear model is by far the most effective way of summarizing such data.
TL;DR: In this article, a linear model approach is proposed for deriving the maximum likelihood estimates of the mother-sib interclass correlation and other parameters from familial data when the families have unequal numbers of offspring.
Abstract: SUMMARY A linear-model approach is proposed for deriving the maximum likelihood estimates of the mother-sib interclass correlation and other parameters from familial data when the families have unequal numbers of offspring. It leads to an algorithm which involves the maximization of an explicit function of a variable for estimating one parameter and direct substitutions for other parameters. The algorithm is believed to be much simpler and more practical than that proposed by Rosner (1979). Some other advantages of the linearmodel approach are also discussed.
TL;DR: The proposed approach is based on maximizing the probabilistic ambiguity between the actual model and the approximate one, and is applicable to general stochastic linear models.
Abstract: Mathematical models, defined by a structure and a set of parameter variation or uncertainty, may, be simplified both by structure reduction and parameter set reduction. First, the approximation of high-order and time varying linear Gaussian models by low-order and time-invariant ones is considered. The proposed approach is based on maximizing the probabilistic ambiguity between the actual model and the approximate one, and is applicable to general stochastic linear models. Reducing a model set, defined on a set of parameter variation or uncertainty, to a single fixed parameter model or a finite model group, is then considered. The set reduction criteria give rise to a min-max optimization problem and a min-max-min problem, which is converted to a constrained min-max problem. The algorithmic solution of the optimization problems is considered in detail, along with several approximation and discretization schemes. The application and the validity, of the proposed approach are examined in view of traditional design considerations by solving numerical examples for several structure and set reduction problems.
TL;DR: In this paper, the authors consider a class of linear models called round robin models which deal specifically with data arising in the interaction of a group of individuals in a round-robin setting and provide information not only about individual differences but also about the reciprocity behavior of the interaction partners.
Abstract: We consider a class of linear models called round robin models which deal specifically with data arising in the interaction of a group of individuals in a round robin setting. Such models provide information not only about individual differences but also about the reciprocity behavior of the interaction partners. We provide a convergent algorithm for computing the maximum likelihood estimates of the variances and covariances associated with these models. Also, we discuss interval estimation of the linear effects, including fixed and random effects. We present a detailed data analysis on a set of speech activity data using these designs.
TL;DR: The recently developed log-linear model technique for the analysis of contingency tables has many potential applications within educational research as discussed by the authors, and the two major models, loglinear model and log linear model, are described in detail in this paper.
Abstract: The recently developed log-linear model technique for the analysis of contingency tables has many potential applications within educational research. This paper describes the two major models, log-...
TL;DR: In this article, it was shown that when all the correlations between residuals are smaller in absolute value than a certain tabulated value, the approximate test based on the first Bonferroni inequality will have size between the nominal value oe and Oh-2°e2.
Abstract: The maximum absolute studentized residual may be used for the detection of a single outlier in a general linear model, but approximations to its distribution are required. Ellenberg (1976, Biometrics 32, 637-645) has proposed the use of the second Bonferroni inequality. It is now shown that, when all the correlations between residuals are smaller in absolute value than a certain tabulated value, the approximate test based on the first Bonferroni inequality will have size between the nominal value oe and Oh-2°e2The results are illustrated by two examples.
TL;DR: In this paper, a self-contained account of minimum disperception linear unbiased estimation of the expectation vector in a linear model with the dispersion matrix belonging to some, rather arbitrary, set of nonnegative definite matrices is given.
Abstract: The paper gives a self-contained account of minimum dispersion linear unbiased estimation of the expectation vector in a linear model with the dispersion matrix belonging to some, rather arbitrary, set of nonnegative definite matrices. The approach to linear estimation in general linear models recommended here is a direct generalization of some ideas and results presented by Rao (1973, 19 74) for the case of a general Gauss-Markov model A new insight into the nature of some estimation problems originaly arising in the context of a general Gauss-Markov model as well as the correspondence of results known in the literature to those obtained in the present paper for general linear models are also given. As preliminary results the theory of projectors defined by Rao (1973) is extended.
1 Jan 1981