Showing papers on "Linear model published in 1974"

Multivariate Statistical Analysis of Glacier Annual Balances

Abstract: A statistical analysis has been made of the annual balances collected during 16 consecutive years at 32 sites on the ablation area of the Glacier de Saint-Sorlin (French Alps). Only 38% of the 32 × 16 balances are known; moreover in 8 cases only the total balance for 2 consecutive years is known, and in one case the balance for 4 consecutive years. A comprehensive study of the errors leads us to assume the following linear model for the annual balance xjt at site j for year t:where αj and βt are parameters depending upon the site and the year respectively, ηjt and η’jt are random errors with a Gaussian distribution and standard errors σ and σ’ respectively. Assuming some known value for σ’2/σ2 = ρ, the parameters αj and βt, their variance–covariance matrix, and the variance covariance matrix of the residuals are estimated in the most general case. The estimators being stable against variations in ρ, the value ρ = o may be assumed; this value docs not conflict with the behaviour of the estimates of the residuals. A test of the linear model derived from Tukey’s non-additivity test is positive. Although a much more general, non-linear model gives a better representation of 13 × 6 balances forming a complete table of data, the linear model with σ ≈ 0.20 m is good enough to be used in theoretical studies or in routine work.

Extreme Vertices Designs for Linear Mixture Models

Abstract: Extreme vertices designs are useful in experimentation with mixtures, particularly when the response can be described by a linear model. An algorithm is proposed for selecting a subset of extreme vertices when the number of candidate vertices is large. This algorithm has been found to produce designs which generally have small trace (X'X)−1, indicating the average variance of the estimated coefficients in the linear model will be small.

Estimation of location parameters from two linear models under normality

Abstract: This paper deals with the problem of estimation of common parameters from two linear models under the assumption of normality. A set of sufficient conditions are obtained under which one can uniformly improve upon the estimates obtained from one model. Uniform improvement on both the models is also considered. We construct estimates which satisfy those conditions and apply these methods to (i) the problem of estimating the common mean of two normal populations and (ii) the problem of recovery of interblock information in incomplete block designs.Exact variances for these estimates and for some other estimates have been evaluated for some impor tant special cases and have been computed for some designs.

Test Statistics for Mixture Models

Abstract: Regression models of the forms proposed by Scheffe and by Becker have been widely and usefully applied to describe the response surfaces of mixture systems. These models do not contain a constant term. It has been common practice to test the statistical significance of these mixture models by the same statistical procedures used for other regression models whose constant term is absent (e.g., because the regression must pass through the origin). In this paper we show that the common practice produces misleading reslllts for mixtures. The mixture models require a different set of F, R 2, and R A 2 statistics. The correct mixture statistics correspond to a physically consistent null hypothesis and are also consistent with the expression of the mixture model in the older “slack-variable” form. An illustrative example is included.

Application of seasonal parametric linear stochastic models to monthly flow data

Abstract: Stochastic linear models are fitted to hydrologic data for two main reasons: to enable forecasts of the data one or more time periods ahead and to enable the generation of sequences of synthetic data. These techniques are of considerable importance to the design and operation of water resource systems. Short sequences of data lead to uncertainties in the estimation of model parameters and to doubts about the appropriateness of particular time series models. A premium is placed on models that are economical in terms of the number of parameters required. One such family of models is multiplicative seasonal autoregressive integrated moving average (Arima) models that have been described by G. E. P. Box and G. M. Jenkins. In this paper we illustrate the process of identifying the particular member of the family that fits logarithms of monthly flows, estimating the parameters, and checking the fit. The seasonal Arima model accounts for the seasonal variability in the monthly means but not the seasonal variability of the monthly standard deviations: for this reason its value is limited. The forecasting of flows one or more months ahead is described with an example.

Application of Linear Statistical Models to Radar Location Techniques

Abstract: The theory of linear statistical models is implemented to obtain an algorithm which accurately locates radar sites. True bearing and navigation data are used as input. The linear model developed is adaptable and allows removal of bearing errors that are nonrandom, or systematic. The model may be written in recursive form and used for real-time applications.

Hypothesis Testing in Linear Models (Eisenhart Model I)

Abstract: The results from an analysis of balanced data are frequently summarized in an analysis of variance (AOV) table. Each sum of squares (SS) in the AOV table is uniquely associated with testing a particular hypothesis in the linear model. These hypotheses are well known and cause no confusion among statisticians as to what is being tested. Results from an analysis of unbalanced data, however, cannot be uniquely summarized in an AOV table, and, consequently, statisticians are often confused about the hypotheses being tested. Some statisticians prefer an orthogonal partitioning of the SS (paralleling the balanced case) as the appropriate analysis; others prefer various forms of nonorthogonal analyses. The purpose of this paper is to show (and, hopefully, clarify) the hypotheses that are being tested in various unbalanced AOV tables.

A mean square error test when stochastic restrictions are used in regression

Abstract: A test is prepared for determining conditions under which stochastic linear prior information, which is incorrect on the average, may improve the parameter estimates for a linear model over conventional sample information estimates, in the sense of having the same or smaller mean square errors for all estimates.

Asymptotic Properties of Multiperiod Control Rules in the Linear Regression Model

Abstract: IK A~ULTIPERIODCONTROL PROBLEMS with unkilown parameters. curreilt decisioiis aKect iiot only current performance, but also the alnoulit of information that is obtained about the uiiknown paraineters. The purpose of this study is to investigate such aspects of m~~l t ipe r iod control in a siinple linear regression model with one unlino\\vn parameter, where t!le illdependent variable is set at certain levels in order to bring the dependent variable to soine desired level. The approach uses the methods and criteria of statistical estimation theory (such as stroiig consisteiicy and eliiciency) to jiivestigate the properties of various control rules. This approacl~ sceins particularly useful in coiitrol problems of this type wliere estimation of unknown parameters plays an important role. Previous i:zvestigatioiis of this type of illultiperiod control problem (Aoki [2]), Zellner [9],aild Prescott [ 5 ] ) have been from a Bayesian point of view. By specifying a loss fiu~~ction, prior distributioris on the paraineters, and a distribution for the random disturbance term, a Bayes co~ltrol rule call be calculated, in principle, with the methods of dy~ialllic programming. However, as these studies have shown, ca1culatio.11 or even characterization of Bayes co~itrol rules has proved quite difficult. The approach of this study is 11011-Bayesian. Thc methods and results should c o m p l e m e ~ ~ t the usual Bayeslail viewpoint in eventually leading to reasoiiabie decisioiis in practical probleins. In Section 2 the nlodel is introduced and two coiltrol rules are defined. In Sectio:~3 we prove that these control rilles converge with probability 1 to tlie value \\vI:icli would be used if the u~ilinown parameter were kliown with certainty. Iil Section 4 we derive tlie asyinptotic distribution of tlie coiitrol ruies aiitl parameter estimates, and in Section 5 we show that these coiitrol rules lead to parameter estilnates whicli have as small an asynlptotic variance as any other control rule in a fairly wide class. 111 particular this nieaiis that control rules which are designed for experimentation d o not give parameter estimates which are any better asy~ilptotically than tile inore simple control rulcs of this paper.

Modeling of DC Motors for Control Applications

Abstract: Three different mathematical models of an armature-controlled dc motor are considered: (i) a precise nonlinear model, (ii) a piecewise linear model, and (iii) a second-order linear model. Experimental results are presented comparing the various models, and a range of applications for each is suggested.

Robust Estimation in the Linear Model

Abstract: sion parameters in the general linear model. We start with a family of M-estimators, and using the observations, we estimate the asymptotic efficiency of each member in the family. Then we choose the estimate in the family with greatest estimated asymptotical efficiency. We prove that this procedure has the same asymptotical efficiency as the member of the family with the greatest asymptotical efficiency for the unknown distribution of the error.

Bates and best quadratic unbiased estimators for variance components and heteroscedastie variances in linear models

Abstract: Let be a linear model with independently - not necessary normally – distribused error components ϵ j and where V(i=1, … p) are known diagonal matrices and the Θ i are unknown scalars (veriance components). Starting from prior distributions with respect to β and Θ BAYES solutions for four elasses of quedratie unblased estimaters for linear functions of the vaciance components are given. They result from solutions of linear equation systems and is general they depend - beside on the experimental design (X,U,V 1,…V p ) -– only on skewness and kurtosis of the ϵ,j 's and on the first two moments of the prior distribution. For special models there oxist solutions depending neither on the prior distribution nor on the distribution of the ϵj 's.

The Average Estimated Variance Criterion for the Selection-of-Variables Problem in General Linear Models

Abstract: The linear models selection-of-variables problem is formulated and the integrated mean square error (IMSE) is discussed as a parametric measure of the “distance” between a true, unknown function, f, and a linear estimating function or “substitute” function, , determined from data. Here where R is a region of interest—a set of x values for which is to be used as a substitute for f, and W(x) is a function which assigns weights to the values of x in R; the weight at x quantifies the importance that (x) be close to f(x). The IMSE, a parameter, cannot be calculated from the data. A statistic which more or less successfully mimics the IMSE in model selection problems is the AEV, defined as: The AEV is introduced, its first two moments are displayed, and for linear functions a simple form of the AEV is derived which uses the second order moment matrix, of R and W: where s 2 is a biased estimate of σ2. The use of the AEV in the linear models selection-ofvariables problem is discussed and illustrated with a proble...

Empirical linear Bayesian decision rules for a sequence of linear models with different eegressor matrices

Abstract: As an approximation of the random parameters in a linear regression model a linear Bayesian decisian rule with restricted minimax property is considered. Since the regression model is assumed t o occur repeatedly (but with different regressor matrix), the unknown para meters of the peior distribution, which are aneded. can be estimated Asymptotie properties of the risk function of the resulding empirical Bayesian decision rnle are is:inverigated.

The Use of Asymptotic Models in Multiple Hormone Assays

Abstract: SUMMARY Hormone assay data are generally analysed by fitting a linear model to a transformation of the variables. An alternative approach which has proved to be very successful for a variety of assays is proposed, and involves the use of a fairly general empirical model. An outline is given of the fitting procedure and two examples presented. The reliability of the standard errors of concentration estimates obtained from the fitted model are checked by means of Monte Carlo methods.

Correcting for unreliability in nonlinear models of attitude change

Abstract: The paper opens with a brief discussion of the problems in testing nonlinear models of attitude change. The regression artifacts produced by unreliability are shown in both the linear and nonlinear case. Classical solutions for the linear case are quickly reviewed. A “new” solution to the linear case is presented and applied to the nonlinear case. It is shown to work well under a broad set of conditions. Regression artifacts in bivariate regression are then discussed. If the predictors are independent, then the univariate correction procedure can be applied to each predictor separately. But if the predictors are correlated, a joint correction procedure must be used. One such procedure is defined and shown to work perfectly in the case of linear regression and reasonably well in a broad set of conditions in which the regression is nonlinear.

Procedures for Estimating Standardized Regression Coefficients From Sample Data

Abstract: This paper is concerned with the role played by the standardized regression coefficients in linear regression analysis. The linear regression model is reparameterized to explicitly contain standardized regression coefficients. Several estimators of these coefficients are considered. It is shown that the usual beta coefficient is a good estimator of the coefficients in the linear regression model with random predictor variables. However, in the linear regression model with nonstochastic predictors, alternative estimators are better than the usual beta coefficient. A sociological application is included in order to display the empirical behavior of the various estimators.

Regression in the Nondifferentiable Bivariate Extreme Models

Abstract: This paper compares general and linear regression for the biextremal and Gumbel bivariate extreme models. The technique is based on the values of the correlation ratios and coefficients. The computations show that linear regression is a good approximation to the general one in both cases.

Psychometric properties of the serial colour word test

Abstract: .— Lower bound reliabilities for the various parameters of two previously suggested models for the serial colour word test—one linear and one quadratic —are derived and estimated. The quadratic model gave mostly very low reliabilities, and the stronger, linear model also gave only a few high reliabilities. Validity was studied by means of discriminant functions (4 groups being used; 3 clinical and 1 normal). The validity of the test was low. The linear model gave the best results under cross validation.