Usual Squares Research Articles

A note on establishing yield-rainfall relationships

Relationships between crop yield or livestock production and associated weather experienced have generally been estimated by way of single equation regression procedures. Where a number of crops and livestock activities are undertaken concurrently in a region, or individual estimates are required for one or more activities across a set of discrete regions, separate estimation of the individual yield relationships is likely to be statistically inefficient, since the residuals from each equation are likely to be correlated. This is because they will in each case include the effects of aspects of weather which are not correlated with the weather indices utilized as independent variables in the regressions. In this situation, provided the independent variables are not all common to all equations, Zellner's (1962) seemingly unrelated estimation procedure provides a statistically efficient means of jointly estimating the set of yield-weather relationships. An application of this methodology which compares it with the usual least squares approach is presented.

Correspondence analysis with least absolute residuals

Three different ways to characterize correspondence analysis as a minimum norm technique are distinguished: an optimal correlation problem, a dual pair of distance approximation problems, and a reciprocal location problem. The consequences of switching from the usual least squares concepts to least absolute residual loss functions are discussed for each of these. The reciprocal location approach gets most attention. Algorithms based on iterative majorization are proposed which are sufficiently general to be adapted to loss functions that are built upon mixtures of least squares and least absolute residuals. A remarkable clustering phenomenon is observed in applications of the new procedure. For the other two approaches no explicit algorithms are given; their development could proceed along the same line.

Selecting a Two-Group Classification Weighting Algorithm.

The cross-validated classification accuracies of three predictor weighting strategies (least squares, ridge regression, and reduced rank) were compared under varying simulated data conditions for the two-group classification problem. Results were somewhat similar to previous findings with multiple regression when absolute rather than relative cross-validation accuracy was the criterion. The alternate weighting strategies performed better than the usual least squares algorithm in many of the data conditions, but the margin by which they surpassed least squares was in many cases quite small. Also, the alternate algorithms were poorer than least squares in some data configurations, and in some cases, by an appreciable margin. Verification of these simulation findings was obtained by documenting agreement with theory, and with an analytic sample-specific procedure that was used with some "real" data sets. The results of the analyses with real data sets yielded results that were similar to those from the simulation. The routine and uncritical use of biased weighting algorithms in classification as advanced by P. J. DiPillo was not supported.

Bayes unbiased estimators of parameters of linear trend with autoregressive errors

The method of least wquares is usually used in a linear regression model $\bold {Y=X\beta+\epsilon}$ for estimating unknown parameters $\bold \beta$. The case when $\epsilon$ is an autoregressive process of the first order and the matrix $\bold X$ corresponds to a linear trend is studied and the Bayes approach is used for estimating the parameters $\bold \beta$. Unbiased Bayes estimators are derived for the case of a small number of observations. These estimators are compared with the locally best unbiased ones and with the usual least squares estimators.

ON THE CONSISTENCY OF LEAST SQUARES ESTIMATORS FOR A THRESHOLD AR(1) MODEL

Abstract. For the SETAR (2; 1,1) modelwhere {at(i)} are i.i.d. random variables with mean 0 and variance σ2(i), i = 1,2, and {at(l)} is independent of {at(2)}, we consider estimators of φ1, φ2 and r which minimize weighted sums of the sum of squares functions for σ2(1) and σ2(2). These include as a special case the usual least squares estimators. It is shown that the usual least squares estimators of φ1, φ2 and r are consistent. If σ2(1) ≠σ2(2) conditions on the weights are found under which the estimators of r and φ1 or φ2 are not consistent.

Errors in Variables: A Problem in Regression and its Solution

It is not uncommon in socioeconomic analysis to measure variables with error, as in a 10% census. The estimation of linear regression coefficients using such ‘errors-in-variables’ models requires modification of the usual ordinary least squares techniques. The underlying theory both for simple and for multiple regression models is explained, and followed up with a numerical example based on a structure plan model of car ownership.

A Compatible System of Growth and Yield Equations for Slash Pine Fitted with Restricted Three-Stage Least Squares

Abstract A system of compatible and interrelating equations (after Clutter 1963) was developed to predict growth and yield in volume, basal area, and dominant height for site-prepared slash pine plantations in the Southeast. Because of logical restrictions on certain parameters and contemporaneous correlations among the variables, the usual unrestricted least squares approach to fitting equations is not appropriate for such a system. The wealth of developments in econometrics for fitting such systems of interrelating equations was reviewed and tested, and restricted three-stage least squares gave the best results. Measures of goodness of fit and precision are nearly as good with restricted three-stage least squares for this system and data as with the conventional unrestricted least squares approach. Methods for placing confidence limits on predicted yields are demonstrated. Forest Sci. 32:185-201.

Vibrational Spectra and Force Constants of Symmetric Tops VL [1] Spin Flip Raman Laser Spectra and Rotational Analysis of (v1 + v2) of CF3Cl and CF3Br

The IR spectra of gaseous CF3Cl and CF3Br with natural isotopic abundances have been investigated in the region 1830-1900 cm-1 with a resolution of 0.01 cm-1 or better using a Q-switched spin-flip Raman laser (SFRL). Both J and K structure of the parallel band (v1 + v2) have been resolved partly for both isotopomers of CF3Cl and CF3Br, respectively. The analysis has been performed by the usual least squares method. Additionally the J clusters of some hot bands have been observed and analyzed.

A Note on Ordinary Least Squares Methods for Two-Stage Sampling

Abstract A class of linear models for two-stage sampling problems is considered. Ordinary least squares estimates are optimal for these models. For some parameters the usual least squares estimates of standard errors are shown to be appropriate.

A Note on Conditional Prediction in the Multivariate Linear Model

SUMMARY In this paper we investigate the effects of errors in the estimation of the covariance structure of a multivariate random vector on the prediction of unobserved components of a further vector conditional on the values of the remaining observed components of the vector. We derive an exact expression for the increase in the mean square error of the usual least squares predictor due to these estimation errors which shows that the increase is substantial for small samples.

A comparison of robust regression techniques for the estimation of weibull parameters

In this paper several alternative robust reqression techniques are compared for estimating parameters of a Weibull distribution . In addition to the usual least squares (L2) and least absolute deviation (L1) methods, a number of one-step reweighting schemes based on the L1residuals are considered. The results of an extensive series of Monte Carlo simulation experiments demonstrate that the Anscmbe reweighting scheme generally produces the best Weibull estimates over the range of sample sizes and parameter values studied.

Robust designs for nearly linear regression

In this paper we seek designs and estimators which are optimal in some sense for multivariate linear regression on cubes and simplexes when the true regression function is unknown. More precisely, we assume that the unknown true regression function is the sum of a linear part plus some contamination orthogonal to the set of all linear functions in the L 2 norm with respect to Lebesgue measure. The contamination is assumed bounded in absolute value and it is shown that the usual designs for multivariate linear regression on cubes and simplices and the usual least squares estimators minimize the supremum over all possible contaminations of the expected mean square error. Additional results for extrapolation and interpolation, among other things, are discussed. For suitable loss functions optimal designs are found to have support on the extreme points of our design space.

A NOTE ON THE MULTIVARIATE LINEAR MODEL WITH CONSTRAINTS ON THE DEPENDENT VECTOR

SummaryIt is shown that the least squares estimators of B and Σ in the multivariate linear model {EYi=X1B, D(Yi) =Σ, 1 ≤i≤n, Y1Yn uncorrelated} subject to the constraints YiM=XiN are just the usual least squares estimators B̂= (X'X)‐1X'Y and ΣC = 1/n(Y‐XB̂)(Y‐XB̂) in the unconstrained model where Σ has full rank. Tests of hypotheses concerning B are discussed for situations in which each Yi has a multivariate normal distribution, and examples of the applicability of the model reviewed.

On the difference between gauss‐markov and least squares estimates

AbstractFor the usual full rank univariate least squares regression model y = XB + e, E(e) = 0, E(ee) = A, the equality of the estimates occurs when B‐B* = (XA−1X)−1XA‐1y‐(XX)−1Xy = 0. A necessary and sufficient condition for this equality is that A has some N ‐ k + 1 roots equal where N is the rank of A and k is the rank of X.

Evaluation of Regression Coefficient Estimates Using α-Acceptability

A framework, called α-acceptability, is proposed for evaluating alternatives to the usual least squares estimates of regression coefficients. An estimate is defined to be α-acceptable if it is in the usual (1 - α) 100% confidence region, Estimates which are α-acceptable for large values of α, such as .99 (corresponding to a 1% confidence region) are viewed as statistically indistinguishable from the usual estimates and, in a sense, simply rounded off. Applications to subset selection, ridge regression, and a class of minimax procedures are discussed and illustrated with an example.

Evaluation of Regression Coefficient Estimates Using α-Acceptability

A framework, called α-acceptability, is proposed for evaluating alternatives to the usual least squares estimates of regression coefficients. An estimate is defined to be α-acceptable if it is in the usual (1 - α) 100% confidence region, Estimates which are α-acceptable for large values of α, such as .99 (corresponding to a 1% confidence region) are viewed as statistically indistinguishable from the usual estimates and, in a sense, simply rounded off. Applications to subset selection, ridge regression, and a class of minimax procedures are discussed and illustrated with an example.

Least squares theory for possibly singular models

AbstractIn a recent paper, Scobey (1975) observed that the usual least squares theory can be applied even when the covariance matrix σ2V of Y in the linear model Y = Xβ + e is singular by choosing the Moore‐Penrose inverse (V+XX′)+ instead of V‐1 when V is nonsingular. This result appears to be wrong. The appropriate treatment of the problem in the singular case is described.

Bioavailability assessment: Methods to estimate total area (AUC 0-∞) and total amount excreted (A e ∞ ) and importance of blood and urine sampling scheme with application to digoxin

Five methods are compared to estimate the total area under the digoxin plasma or serum concentrationtime curve (AUC0-∞) after a single dose of drug. To obtain accurate estimates of AUC0-∞, data required are concentrations at a sufficient number of sampling times to define adequately the concentration-time curve prior to the log-linear phase, and at least three, but preferably four or more equally spaced points in the terminal loglinear phase. One method (designated Method I) requires a digital computer; another (Method III) is the classical method (these two methods do not require equally spaced points in the loglinear phase). Method IIA is the accelerated convergence method of Amidon et al.; Methods IIB and IIC are modifications of this method, but incorporate usual and orthogonal least squares, respectively, which make them more accurate with real (noisy) data. Methods I and IICgave very comparable estimates of AUC0-∞. Results indicate that digoxin administered orally in aqueous solution was completely (100%) absorbed when bioavailability estimates were based on oral and intravenous AUC0-∞ estimates and the actual doses, whereas formerly only about 80% absorption was reported, based on areas, under plasma concentration curves which were truncated at 96 hr. It is shown that the sampling scheme of blood can produce biased apparent bioavailability estimates when areas under truncated curves are employed, but an appropriate sampling scheme and application of method IIyield accurate bioavailability estimates. This is important particularly in those bioavailability studies where one is attempting to determine the appropriate label dose for a new “fastrelease” digoxin preparation relative to the label dose and bioavailability of currently marketed tablets. It is shown that the magnitudes and variability of apparent elimination rate constants and halflives of digoxin, estimated from urinary excretion data by the σ− method, depend on which value of Ae∞ is used. The formerly reported greater interindividual variability of AUC data compared to At data for digoxin is explained in that the AUCs, but not the Ae,'s, involve the renal clearance, which exhibits considerable inter- and intraindividual variation.

A Bayesian approach to prediction using polynomials

SUMMARY We have an unknown function h(x) which we want to estimate within a finite interval. The observed values of h(x) are independent observations of a random variable y whose mean is to be approximated by a polynomial of unknown degree. The problem of estimating h(x) then translates into that of predicting y. We assume that the mean of y is a polynomial of an arbitrarily large degree and derive a prior distribution for its coefficients which expresses the belief that these coefficients will tend to decrease in absolute value as the power of x increases. A prior-posterior analysis for the coefficients is carried out from which we obtain modal estimates of them. We derive the predictive distribution of y for the case when all parameters other than the mean are known. Two examples comparing the performance of this procedure with some of the usual least squares ones are presented.

A discrete least squares method

We consider a discrete least squares approximation to the solution of a two-point boundary value problem for a 2mth order elliptic operator. We describe the approximation space of piecewise polynomials and devise a Gaussian quadrature rule that is suitable for replacing the integrals in the usual least squares method. We then show that if the quadrature rule is of sufficient accuracy, the optimal order of convergence is obtained.