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Under certain assumptions the stage-discharge relationship of a channel cross-section can be approximated by a logarithmic relationship. Observational pairs of stage and discharge plotted on log-log paper often cluster around a straight line and this suggests that the assumptions involved are often approximately satisfied. In such cases the parameters of the logarithmic relationship are usually estimated graphically from the position and slope of the straight line on the log-log paper. In this paper principles and methods are outlined for the estimation of the parameters with estimates of their standard error, via regression analysis. Because the water level of zero flows is usually one of the unknown parameters, the regression is non-linear and least squares optimal estimates can be obtained by a step-by-step approximation. The variances of the parameter estimates can be obtained from the dispersion matrix of the joint distribution of the least squares estimators via the likelihood function. An ...

In a previous paper [6] the covariances of least-squares estimates of regression coefficients and the expected value of the estimate of residual variance were investigated when the errors are assumed to be correlated. In this paper we will investigate the distribution of the usual test statistics for regression coefficients under the same assumptions. Applications of the theory to the cases of testing a single sample mean, the difference between the means of two samples, the coefficients in a linear trend and in regression on trigonometric functions will be discussed in some detail under an assumed covariance matrix for errors.

In the statistical literature, Booth et al. (Ref. 1) and Hartley (Ref. 2) developed modifications of the well-known Gauss-Newton method of iterative solution and applied it to estimate a set of parameters for nonlinear regression problems by least squares. To ensure convergence of his method, Hartley restricts its applicability to problems that satisfy his assumptions. Herein, we describe a method that removes two of Hartley's three assumptions, thereby making it applicable to those problems for which his method fails to converge. The method we propose is based upon an algorithm due to Davidon (Ref. 3). It is an iterative descent method whose rate of convergence is quadratic in the limit. It was later modified by Fletcher and Powell (Ref. 4), and has been used for locating an unconstrained local minimum of a function of several variables. Fletcher and Powell's account of Davidon's method has been found to be very useful when first derivatives of the function are available. However, in realistic situations it frequently is practically impossible to calculate first derivatives. Therefore, the method we describe is a modification of the basic Fletcher and Powell method due to McGill (Ref. 5) and Taylor (Ref. 6) and includes the case for which the gradient is not given analytically. Our concern is with the application and introduction of the modified Davidon method (MDM) to nonlinear regression problems of the type discussed by Hartley. We have omitted all proofs concerning stability and rate of convergence of the method and refer the reader to the excellent paper of Fletcher and Powell for such proofs. Included is a description of the MDM, a description of ways of terminating the iterations, and an illustration of the method with the numerical example Hartley used in his paper. We show the results for both the case of analytic and approximate gradient. Also included are results obtained in applying both methods to a second example, which involves the estimation of six parameters for an exponential regression function. For this example, it is shown that the method described in this note converges when Hartley's method fails to converge.

The paper presents a method for resolving additive mixtures of overlapping curves by combining nonlinear regression and principal component analysis. The method can be applied to spectroscopy, chromatography, etc. The method makes use of the postulated chemical reaction, and allows one to check the reaction and estimate chemical rate and equilibrium constants.

This paper gives alternative sufficient conditions for the least squares estimates to be consistent in the case of nonlinear regression, i.e., without the assumption of linearity of g with respect to the parameters.

Before the advent of nonlinear optimization algorithms, the solution of chemical reaction or heat transfer equations was attempted primarily by trial and error or linear approximation methods. In the method illustrated by the two examples in this paper, any set of nonlinear equations for a chemical or physical process are replaced by a single equation. The variables of interest are in the resulting functional equation treated as parameters whose optimum values are determined by a nonlinear estimation technique. These values are considered as nonlinear regression coefficients which are simultaneously adjusted by an optimization algorithm. The method gives an estimate of the standard deviation in the determination of the value of each variable and provides information concerning the errors associated with their interaction or pairwise correlations. Validity of the results is confirmed by comparison with solutions independently obtained through a matrix technique developed by Wu (1). For each of the examples the magnitude of computed residual is provided as a measure of the precision of the solution set. The general method is applicable to a very wide class of physical and chemical problems described by one or more nonlinear equations. Avant l'avenement des algorithmes de rendement optimum et non-lineaire, on cherchait a trouver la solution des equations des reactions chimiques ou du transfert de la chaleur surtout par les methodes d'approximation successive et lineaire. Dans la methode illustree par deux exemples dans ce travail, on remplace n'importe quelle serie d'equations non-lineaires relatives a un procede chimique ou physique par une seule. On traite les variables interessantes dans l'equation fonctionnelle qui en resulte comme des parametres dont les valeurs optimales sont determinees par une methode d'evaluation non-lineaire. On considere ces valeurs comme des coefficients de regression non-lineaire lesquels sont regles simultanement par un algorithme de rendement optimum. La methode fournit une estimation de la deviation nominale dans la determination de la valeur de chaque variable ainsi que des renseignements sur les erreurs qui accompagnent leur action mutuell ou les correlations en paire. On confirme la validite des resultats en les comparant avec les solutions qu'on a obtenues independamment par la methode a base de matrice mise au point par Wu 1. Dans le cas de chacun des exemples, la grandeur du residu calcule sert de mesure de la precision de la serie de solutions. La methode generale s'applique a un grand nombre de problemes physiques et chimiques decrits dans une ou plusieurs equations non-lineaires.

The sequential procedure for planning of experiments aiming at optimum discrimination between rival models, proposed by Box and collaborators, was applied to the isomerization of n - pentane data of Hosten and Froment. Subsequently the sequential design procedure for optimum parameter estimation introduced by Box and collaborators is illustrated. The paper also compares parameter estimates obtained by minimizing the residual sum of squares of space time and of conversion. The first method implies linear, the second non linear regression.

In this paper a method of estimating the parameters of a set of regression equations is reported which involves application of Aitken's generalized least-squares [1] to the whole system of equations. Under conditions generally encountered in practice, it is found that the regression coefficient estimators so obtained are at least asymptotically more efficient than those obtained by an equation-by-equation application of least squares. This gain in efficiency can be quite large if “independent” variables in different equations are not highly correlated and if disturbance terms in different equations are highly correlated. Further, tests of the hypothesis that all regression equation coefficient vectors are equal, based on “micro” and “macro” data, are described. If this hypothesis is accepted, there will be no aggregation bias. Finally, the estimation procedure and the “micro-test” for aggregation bias are applied in the analysis of annual investment data, 1935–1954, for two firms.

The material presented furnishes an extension of regression analysis in two respects. First, a type of nonlinear regression function is developed that seems to (1) have substantial curve-fitting flexibility, (2) be satisfactorily determined from an acceptably small number of observations, (3) permit isolation of important combinations of effects, and (4) be computationally manageable. Second, in conjunction with this general type of regression function, a probability model is developed that is applicable under exceedingly general conditions. Namely, each of the independent observations utilized can be from a different statistical population and the shapes of these populations are not necessarily restricted or related in any specific manner. An important aspect of the model is selection of the kind of “average” that the regression represents Estimates, tests, and confidence intervals are developed for investigating the regression coefficients. The methods and results presented are especially useful for Monte Carlo type simulations that are performed by use of a computer. This paper represents a report of research in progress, since many additional results should be obtainable on the basis of the concepts and methods that are presented.

The paper is concerned with parametric models for populations of curves; i.e. models of the form yi (Z) = f(θ i ; x) + error, i = I, 2, …, n. The shape invariant model f(θ i ; x) = θ0i + θ1i g([x – θ2i /θ3i ) is introduced. If the function g(x) is known, then the θ i may be estimated by nonlinear regression. If g(x) is unknown, then the authors propose an iterative technique for simultaneous determination of the best g(x) and θ i . Generalizations of the shape invariant model to curve resolution are also discussed. Several applications of the method are also presented.

Watson (1955) investigated the performance of a regression analysis based on the assumption that the error covariance matrix is o2y when it is, in fact, o2x. In the present paper Watson's results regarding the effects of this type of specification error on the bias of estimators of variance are generalized. In particular, we give, for arbitrary design matrices of full rank, attainable bounds for the bias of the least-squares estimator of the variance of arbitrary linear functions of the estimated regression coefficients.

We are given a set of $N$ responses $Y_t$ which have arisen from a nonlinear regression model \begin{equation*}\tag{(1.1)}Y_t = f(x_t, \theta) + e_t; \quad t = 1, 2, \cdots, N.\end{equation*} Here $x_t$ denotes the $t$th fixed input vector of $k$ elements giving rise to $Y_t$, whilst $\theta$ is an $m$-element unknown parameter vector with elements $\theta_i$ and the $e_t$ are a set of $N$ independent error residuals from $N(0, \sigma^2)$ with $\sigma^2$ unknown. The expectations of the $Y_t$, are therefore the functions $f(x_t, \theta)$ which will be assumed to satisfy certain regularity conditions. The problem is to estimate $\theta$ notably by least squares. In this paper we shall develop an iterative method of solution of the least squares equations which has the following properties: (a) the computational procedure is convergent for finite $N$; (b) the resulting estimators are asymptotically $100{\tt\#}$ efficient as $N \rightarrow \infty$. In Sections 2-4 we give a survey of our results leaving the mathematical proofs to Sections 5-7 whilst in Section 8 we illustrate our method with an example. Although our theoretical development is oriented towards our specific goals certain results are proved in a somewhat more general form. Some of our theory will be seen to correspond to well known theorems on stochastic limits which have to be reproved because of certain modifications which we require.

A great many special methods for the statistical analysis of sensitivity or quantal response data have been developed during the past century. This paper demonstrates that many of these techniques can be considered in the light of non-linear regression methods which have been made more tolerable in recent years with the prevalence of high-speed computers.

For the regression model $Y_{\nu i} = \alpha + \beta C_{\nu i} + \epsilon_{\nu i}, i = 1, \cdots, N_\nu$, where the $\epsilon_{\nu i}$ are independent and identically distributed random variables (iidrv), optimum rank order tests for the hypothesis that $\beta = 0$ are due to Hoeffding (1950), Terry (1952) and Hajek (1962), among others. In the present paper, the theory is extended to the problem of testing the homogeneity of the regression coefficients from $k(\geqq 2)$ independent samples. Allied efficiency results are also presented.

1. Summary. This paper studies the problem of how to space observations in regression so as to minimize the variance of an estimate of the regression function value at an arbitrary point in the domain of observations. Necessary and sufficient conditions are obtained for such a design, called a minimax design, in two dimensional polynomial regression of the type in which the regression function possesses a product structure. Such conditions are also obtained for minimax designs in one dimensional trigonometric and two dimensional spherical harmonics regression. Particular designs of the latter type are constructed. 2. Introduction. Let f1(x), * * * , fk(x) be a set of linearly independent continuous functions defined on a bounded compact domain X and let yx denote a random variable associated with x whose mean is given by the regression value

SUMMARY In this paper simulation techniques are used to evaluate the use of linear programming in regression analysis. The experiments demonstrate that, in certain cases, minimizing the sum of the absolute values of the deviations (L1 norm) is preferable to the Least Squares criterion. No significant bias was found in the L1 norm estimates.