Abstract: In a heteroscedastic linear model, it is known that if the variances are a parametric function of the design, then one can construct an estimate of the regression parameter which is asymptotically equivalent to the weighted least squares estimate with known variances. We show that the same is true when the only thing known about the variances is that they are determined by an unknown but smooth function of the design or the mean response.

Abstract: In this paper we analyze the solutions of linear econometric models with rational expectations. More precisely, we describe in detail the set of all the solutions; in particular this set is shown to be much larger than the sets previously considered. We also study various criteria of selection in this set of solutions and we examine to what extent these criteria redtiuce the set of the solutions.

Abstract: In analogy to analysis of variance for linear models, analyses of dispersion for loglinear models for multinomial responses are constructed. The analyses, which are based on the entropy and concentration measures, are used to construct tests of independence and measures of association.

Abstract: A theorem is proved which demonstrates that an earlier derived minimum description length estimation criterion is capable of distinguishing between structures in linear models for vector processes. A fairly simple algorithm is described for the estimation of the best model, including its structure and the number of its parameters.

Abstract: Monte Carlo simulation has been used to test the robustness of three methods of regression in the comparison of analytical accuracy. The methods used were simple linear regression, weighted linear regression and a new method based on maximum likelihood estimation. The new method is capable of unbiased estimation with different heteroscedastic variance of both the y-values and the x-values, and is the most generally useful. However, even simple linear regression can estimate linear biases accurately as long as elementary safeguards are applied. The simulations realistically covered a wide range of circumstances likely to be encountered in analytical practice.

Abstract: The probability of ruin is examined in a model where the annual gains of an insurance company are dependent random variables. The model used is the linear model (well known in time-series analysis) which includes the autoregressive model and the moving average model as special cases. It is also shown that a certain credibility model can be interpreted as a first-order model of the mixed type.

Abstract: 1 Some properties of basic statistical procedures.- 1.1 Problems of statistics.- 1.2 The t, X2 and F procedures.- 1.3 Standard assumptions and their plausibility.- 1.4 Tests of normality.- 1.5 Moments of $$\bar{x}$$ and s2.- 1.6 The effect of skewness and kurtosis on the t-test.- 1.7 The effect of skewness and kurtosis on inferences about variances.- 1.8 The effect of serial correlation.- 1.9 The effect of unequal variances on the two-sample t-test.- 1.10 Discussion.- Further reading.- 2 Regression and the linear model.- 2.1 Linear models.- 2.2 The method of least squares.- 2.3 Properties of the estimators and sums of squares.- 2.4 Further analysis of Example 2.1.- 2.5 The regressions of y on x and of x on y.- 2.6 Two regressor variables.- 2.7 Discussion.- 3 Statistical models and statistical inference.- 3.1 Parametric inference.- 3.2 Point estimates.- 3.3 The likelihood function.- 3.4 The method of maximum likelihood.- 3.5 The Cramer - Rao inequality.- 3.6 Sufficiency.- 3.7 The multivariate normal distribution.- 3.8 Proof of the Cramer - Rao inequality.- Further reading.- 4 Properties of the method of maximum likelihood.- 4.1 Introduction.- 4.2 Formal statements of main properties.- 4.3 Practical aspects - one-parameter case.- 4.4 Practical aspects - multiparameter case.- 4.5 Other methods of estimation.- 5 The method of least squares.- 5.1 Basic model.- 5.2 Properties of the method.- 5.3 Properties of residuals.- 5.4 Properties of sums of squares.- 5.5 Application to multiple regression.- Further reading.- 6 Multiple regression: Further analysis and interpretation.- 6.1 Testing the significance of subsets of explanatory variables.- 6.2 Application of the extra sum-of-squares principle to multiple regression.- 6.3 Problems of interpretation.- 6.4 Relationships between sums of squares.- 6.5 Departures from assumptions.- 6.6 Predictions from regression.- 6.7 Strategies for multiple regression analysis.- 6.8 Practical details.- Further reading on practical points.- 7 Polynomial regression.- 7.1 Introduction.- 7.2 General theory.- 7.3 Derivation of the polynomials.- 7.4 Tables of orthogonal polynomials.- 7.5 An illustrative example.- 8 The use of transformations.- 8.1 Introduction.- 8.2 One explanatory variable.- 8.3 Transformations for homogeneity of variance.- 8.4 An example.- 8.5 The Box-Cox transformation.- 8.6 Transformations of regressor variables.- 8.7 Application to bioassay data.- Further reading.- 9 Correlation.- 9.1 Definition and examples.- 9.2 Correlation or regression?.- 9.3 Estimation of ?.- 9.4 Results on the distribution of R.- 9.5 Confidence intervals and hypothesis tests for ?.- 9.6 Relationship with regression.- 9.7 Partial correlation.- 9.8 The multiple correlation coefficient.- Further reading.- 10 The analysis of variance.- 10.1 An example.- 10.2 Generalized inverses.- 10.3 Least squares using generalized inverses.- 10.4 One-way classification analysis of variance.- 10.5 A discussion of Example 10.1.- 10.6 Two-way classification.- 10.7 A discussion of Example 10.2.- 10.8 General method for analysis of variance.- Further reading.- 11 Designs with regressions in the treatment effects.- 11.1 One-way analysis.- 11.2 Parallel regressions.- 11.3 The two-way analysis.- 12 An analysis of data on trees.- 12.1 The data.- 12.2 Regression analyses.- 12.3 The analysis of covariance.- 12.4 Residuals.- 13 The analysis of variance: Subsidiary analyses.- 13.1 Multiple comparisons: Introduction.- 13.2 Multiple comparisons: Various techniques.- 13.3 Departures from underlying assumptions.- 13.4 Tests for heteroscedasticity.- 13.5 Residuals and outliers.- 13.6 Some points of experimental design: General points.- 13.7 Some points of experimental design: Randomized blocks.- Further reading on experimental design.- 14 Components of variance.- 14.1 Components of variance.- 14.2 Components of variance: Follow-up analysis.- 14.3 Nested classifications.- 14.4 Outline analysis of Example 14.3.- 14.5 Nested classifications: Finite population model.- 14.6 Sampling from finite populations.- 14.7 Nested classifications with unequal numbers.- Further reading.- 15 Crossed classifications.- 15.1 Crossed classifications and interactions.- 15.2 More about interactions.- 15.3 Analysis of a two-way equally replicated design.- 15.4 An analysis of Example 15.1.- 15.5 Unit errors.- 15.6 Random-effects models.- 15.7 Analysis of a two-way unequally replicated design.- Further reading.- 16 Further analysis of variance.- 16.1 Three-way crossed classification.- 16.2 An analysis of Example 16.1.- Further reading.- 17 The generalized linear model.- 17.1 Introduction.- 17.2 The maximum likelihood ratio test.- 17.3 The family of probability distributions permitted.- 17.4 The generalized linear model.- 17.5 The analysis of deviance.- 17.6 Illustration using the radiation experiment data.- Further reading.- References.

Abstract: SUMMARY This paper examines modelling a single outlier in the normal theory fixed effects linear model as arising from an unknown observation with inflated variance. The maximum likelihood estimates are characterized in terms of standard least squares statistics. The estimated position of the outlier does not necessarily agree with the estimated position under the usual mean slippage outlier model, and an example where they differ is presented. A sufficient and common condition for agreement is given. leads to considering the Studentized residuals, since, under normality and the possibility of each observation being the outlier, the maximum likelihood estimate for the position of the possible outlier corresponds to the case with the largest absolute Studentized residual. Also, the likelihood ratio test statistic for the presence of an outlier in this model is a monotonic function of the largest absolute Studentized residual (Srikantan, 1961; Tietjen et al., 1973; Ellenberg, 1976). Intuitively, one might expect similar results to be true whenever the effect of a single outlier is modelled identically for all observations, regardless of values of the independent variables or fixed effects. We shall see that this is not necessarily so. We consider here a single outlier model that assumes an outlier arises from an error term with an inflated variance (Box and Tiao, 1968, consider this model from a Bayesian point of view). In Section 2 the model is defined and shown to be, after a parameter transformation, a special case of Harville's general linear model (Harville, 1977). The joint maximum likelihood estimates (MLE's) are characterized in Section 3, and an example where the estimated outlier does not correspond to the observation with the largest absolute Studentized residual is presented in Section 4. In the final section several issues pertinent to the modelling of outliers are discussed in terms of this model and the mean slippage model. Our intent here is not to propose this model as a common replacement for the mean slippage model, but rather to examine the performance of the mean slippage model under a plausible alternative.

Abstract: Statistical texts differ in the ways they test the significance of coefficients of lower-order terms in polynomial regression models. One reason for this difference is probably the concern of some authors that t ratios for the regression coefficients of lower-order terms may change as a result of linear scale transformations of the independent variables. Empiricists generally report tests of these terms and attempt to interpret their meanings. This paper succinctly demonstrates how linear transformations of polynomial regression models affect tests of statistical significance for individual parameters.

Abstract: The present paper is concerned with optimal estimation of the 1st and 2nd order structural moments appearing in credibility formulas. In a recent paper De Vylder has treated the problem in the case of multinormal conditional distributions under quite restrictive assumptions. He minimizes, within a certain restricted class of unbiased estimators, the variance (or the sum of variances if the estimand is a matrix) and next replaces all structural moments (up to fourth order) in the solution by estimates based on the data. This paper is an attempt to simplify the method and extend it so as to make it applicable in more general situations. By suitable choice of a (sufficient) set of statistics and a suitable parametrization, the powerful theory of estimation in linear models can be employed, which makes cumbersome minimization procedures superfluous. The theory is applied to the cases with binomial. Poisson, compound Poisson, and multinormal conditional distributions. Some simulation studies have been performed to assess the performance of the estimators.

Abstract: The problem of linear model structure identification for multivariate time series or multiple input-output models is presented and solved The identification is obtained using canonical correlations to determine model order The equivalence between state-space model structure and multivariate autoregressive moving average with exogenous inputs (ARMAX) models is presented The class of models open to analysis includes rainfall-runoff models, multivariate streamflow models, and time invariant state-space models used in Kaiman filtering Examples include a rainfall-runoff model using three precipitation inputs, a four-site monthly streamflow model, and a four-season streamflow model

Abstract: Let Y be distributed symmetrically about Xβ. Natural generalizations of odd location statistics, say T‘Y’, and even location-free statistics, say W‘Y’, that were used by Hogg ‘1960, 1967)’ are introduced. We show that T‘Y’ is distributed symmetrically about β and thus E[T‘Y’] = β and that each element of T‘Y’ is uncorrelated with each element of W‘Y’. Applications of this result are made to R-estiraators and the result is extended to a multivariate linear model situation.

Abstract: Standard methods for nonlinear equations and unconstrained minimization base each iteration on a linear or quadratic model of the objective function, respectively. Recently, methods using two generalizations of the standard models have been proposed for these problems. Conic methods for unconstrained minimization use a model that is the ratio of a quadratic function divided by the square of a linear function. Tensor methods for nonlinear equations augment the standard linear model with a simple second order term. This paper surveys the research to date on methods for unconstrained minimization and nonlinear equations that use conic and tensor models. It begins with a brief summary of the standard methods, so that the paper is essentially selfcontained.

Abstract: Davidson and MacKinnon (1981) proposed a simple procedure for testing the specification of a non-linear regression model against the evidence provided by a non-nested alternative. We extend their results in several directions. First, we relax a number of assumptions of the previous paper: We admit the possibility that the nonlinear regression functions may depend on lagged dependent variables, and we do not require that the error terms be normally distributed. Second, we show how the earlier procedure may straightforwardly be generalized to the case where the two non-nested models involve different transformations of the dependent variable. Finally, we propose a simple procedure for testing non-nested linear regression models which have endogenous variables on the right-hand side, and have therefore been estimated by two-stage least squares.

Abstract: The time‐resolved fluorescence relaxation obtained by a pulse method can be described by a convolution product of the measured excitation profile and the fluorescence response of the sample. In this paper, the behavior of this convolution model, with respect to the parameters, is examined. The model happens to be nearly linear in the neighborhood of the least‐squares estimates, according to an adapted nonlinearity measure. As a consequence, the lack of fit‐test and the extra sum of squares‐test from the theory of linear models can be applied to decide how many components should be included in the fluorescence response function. The proposed method can be easily extended to other areas involving exponential relaxation.

Abstract: The relative accuracies of various estimators of partial regression coefficients are investigated for the case of two Independent variables x1 and x2 with, randomly missing values on x2 only. The estimators are studied.in the context of the usual linear model yi = β0 + β1x1 i, + β2x2 i + ci. where the ci are i.i.d.with mean 0 and variance α2. The mean square errors of two estimators% the piecewise estimator and the linear prediction estimator, are derived for both β1. and β2 and compared with the nuaan square error of ihe complete-case estimator* To further compare the estimators,, a Monte Carlo study of oliser¥atio»iis generated from a trivariate normal distribution is performed. The study supports a general preference for the method of maximum likelihood in estimating both β1 and β2 under this model

Abstract: Parameters which characterize geotechnical systems and are representative of the behaviour of soil or rock masses are often known with a high degree of uncertainty A way of reducing uncertainties and thus improving the mathematical models for analysis and design purposes is provided by a systematic adjustment of parameters so that theoretical predictions through the model match observational data This identification or “inverse” problem frequently implies recourse to techniques of mathematical optimization and particularly of mathematical programming This paper concerns the role played in this context by direct search methods, least squares procedures and quadratic and nonlinear programming for identifying parameters in purely numerical models, linear models and linear complementarity models, respectively A procedure of statistical identification is briefly mentioned at the end The scope is not to survey the field but merely to elucidate some aspects and potentialities of identification methods in geomechanics on the basis of a variety of recent results on particular problems