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.

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.

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.

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.

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.

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.

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.

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.

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...

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.

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.

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.

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.

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.

Abstract: Estimation of parameters is considered for several cases involving correlated errors. The cases include first and second order cumulative errors and a more general first order case. Estimators for first order cumulative errors are tabulated for five simple linear models. More general estimators for linear models are given in matrix form and it is demonstrated using maximum likelihood that many cases can be written simply in the form of differences. Expressions are also given that can be used to estimate the correlation coefficient and the error variance. Two examples are given to illustrate the new results.

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.

Abstract: This article considers the relation between two discrete stochastic models used in population dynamics—namely the additive errors model and the branching process model. It is shown that every Galton-Watson process can be generated by an additive errors model with a suitably specified error process. Using this result, formulas are derived for the asymptotic behavior of the branching process.

Abstract: Adopting a linear nonstationary model for the surface runoff system gives a better fit to a set of observed direct surface runoff hydrographs than adopting a stationary linear model. The kernel of this model, which is a function of two time variables, can be considered to be an assembly of impulse response functions for unit pulses acting at various time intervals after the beginning of the storm. Each of these impulse response functions must have the properties of an ordinary instantaneous unit hydrograph. The kernel function is evaluated by an optimization procedure that seeks to minimize the sum of squared deviations between the observed and computed hydrographs for a number of independent storms, subject to given constraints. The procedure is based on a discretization scheme for the functions involved. The set of equations for the unknown kernel values can be partitioned into independent subsets, each of which is solved individually. The complete equation set need be considered only when area constraints of the kernel function are imposed.