Squares Estimator Of Regression Coefficients Research Articles

PMC Theorems on PCR–Ridge Class Estimators

After presenting the basic linear model and some definitions, the PMC superiority for a class of biased estimators is discussed. One objective of this paper is to develop such a methodology for the case when the squared statistical distance between the estimator and the parameter is used as the loss function, and that the estimators to be compared are linear combinations of common statistics having a multivariate normal distribution. To demonstrate the usefulness of the methodology, it is applied for the comparison of $$r-k$$ class estimators with unbiased least squares estimator of regression coefficients.

Influence measure based on probabilistic behavior of regression estimators

An influence measure for investigating the influence of deleting an observation in linear regression is proposed based on geometric thoughts of the sampling distribution of the distance between two estimators of regression coefficients computed with and without a single specific observation. The covariance matrix of the above sampling distribution plays a key role in deriving the influence measure. It turns out that geometrically, this distance is distributed entirely along the axis associated with the nonnull eigenvalue of the covariance matrix. The deviation of the regression coefficients computed without an observation from the regression coefficients computed with the full data is reflected in the eigenvalue of the covariance matrix which can be used for investigating the influence. The distance is normalized using the associated covariance matrix and this normalized distance turns out to be the square of internally studentized residuals. Illustrative examples for showing the effectiveness of the influence measure proposed here are given. In judging the influence of observations on the least squares estimates of regression coefficients, Cook's distance does not work well for one example and therefore we should be cautious about a blind use of the Cook's distance.

Comparative study of response surface designs with errors-in-variables model

This paper investigates the scaled prediction variances in the errors-in-variables model and compares the performance with those in classic model of response surface designs for three factors. The ordinary least squares estimators of regression coefficients are derived from a second-order response surface model with errors in variables. Three performance criteria are proposed. The first is the difference between the empirical mean of maximum value of scaled prediction variance with errors and the maximum value of scaled prediction variance without errors. The second is the mean squared deviation from the mean of simulated maximum scaled prediction variance with errors. The last performance measure is the mean squared scaled prediction variance change with and without errors. In the simulations, 1 000 random samples were performed following three factors with 20 experimental runs for central composite designs and 15 for Box-Behnken design. The independent variables are coded variables in these designs. Comparative results show that for the low level errors in variables, central composite face-centered design is optimal; otherwise, Box-Behnken design has a relatively better performance.

Ridge regression technique to determine the environmental influences on tef (Eragrostis tef) grain yield

Tef grain yield is dependent on a number of component characters such as plant height and panicle length. These characters and consequently yield are governed by a large number of factors including environmental factors. The objective of this paper was to determine the environmental influences on tef grain yield. The effects of eleven environmental variables on tef yield were studied using least squares and ridge regression analyses. The results revealed that the least squares estimates of regression coefficients for seven environmental variables did not give a correct indication of the influence of the variables on tef yield but that the estimates from ridge regression were stable. In addition, the ridge regression model has a lower mean square error. Tef grain yield is positively correlated to rainfall, average monthly minimum temperatures, soils dominated by silt with adequate quantities of available nitrogen while it is negatively influenced by average monthly maximum temperatures and clay soils. The results obtained from this study are in accordance with previous literature about the effects of environmental factors on yield of tef cultivars.

Secure analysis of distributed chemical databases without data integration

We present a method for performing statistically valid linear regressions on the union of distributed chemical databases that preserves confidentiality of those databases. The method employs secure multi-party computation to share local sufficient statistics necessary to compute least squares estimators of regression coefficients, error variances and other quantities of interest. We illustrate our method with an example containing four companies' rather different databases.

Secure Regression on Distributed Databases

This article presents several methods for performing linear regression on the union of distributed databases that preserve, to varying degrees, confidentiality of those databases. Such methods can be used by federal or state statistical agencies to share information from their individual databases, or to make such information available to others. Secure data integration, which provides the lowest level of protection, actually integrates the databases, but in a manner that no database owner can determine the origin of any records other than its own. Regression, associated diagnostics, or any other analysis then can be performed on the integrated data. Secure multiparty computation, based on shared local statistics effects computations necessary to compute least squares estimators of regression coefficients and error variances by means of analogous local computations that are combined additively using the secure summation protocol. We also provide two approaches to model diagnostics in this setting, one using shared residual statistics and the other using secure integration of synthetic residuals.

Linear Least Squares Estimation of Regression Models for Two-Dimensional Random Fields

We consider the problem of estimating regression models of two-dimensional random fields. Asymptotic properties of the least squares estimator of the linear regression coefficients are studied for the case where the disturbance is a homogeneous random field with an absolutely continuous spectral distribution and a positive and piecewise continuous spectral density. We obtain necessary and sufficient conditions on the regression sequences such that a linear estimator of the regression coefficients is asymptotically unbiased and mean square consistent. For such regression sequences the asymptotic covariance matrix of the linear least squares estimator of the regression coefficients is derived.

Residuals and Influence in the Multivariate Linear Model

Regression diagnostics for the multivariate linear model are developed along the lines of the theory for the linear model with univariate response. Internally and externally studentised forms of residuals are given and their distributions found. Distance measures suitable for the assessment of the influence of particular cases on the estimated regression coefficients are considered. The examination of residuals and influence statistics is of great importance in assessing a regression model. Cook & Weisberg (1982) provide an extensive discussion of relevant methods for the linear model with a single response variable. The purpose of the present note is to apply these ideas to the multivariate linear regression problem. Although ordinary least squares estimates of regression coefficients are the same in the multivariate and univariate analyses, there are obvious reasons for carrying out the multivariate analysis to consider simultaneously the different re- sponse variables. One is the possibility that the residual for one response variable in a particular case may not seem to be out of the ordinary in relation to other residuals for that response, but only in relation to the residuals for other responses on the same case. Another is that we may be interested in specifically multivariate aspects of the data. For example, in the problem that prompted this investigation, the main item of interest was the matrix of inter-correlations between five indicators of pollution from sampling stations in the Aegean Sea. This was calculated as the matrix of correlations between the residuals from the regressions of the indicators on covariates including temperature and pH of the seawater. Correlations are particularly vulnerable to distortion by outlying values (Gnanadesikan & Kettenring, 1972), so examination of the multivariate residuals to protect against this was essential. In the following two sections, multivariate residuals and influence measures are presented. Section 4 outlines an application to illustrate the usefulness of the method-

On a conjecture of Hoerl and Kennard on a property of least squares estimators of regression coefficients

A conjecture of Hoerl and Kennard on a property of least squares estimators is established in a general setup.

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.

Consistent estimation of a regression with errors in the variables

As is well known, least squares estimates of regression coefficients are inconsistent if the variables are measured with random errors. In the classical case of known variances and covariances for these error variables, consistent estimates can be derived. It is shown that these estimators generally have a joint asymptotic normal distribution, the covariance matrix of which is derived. No use is made of normality assumptions, but knowledge of the third and fourth moments of error variables is utilized.