Simple Regression Problem Research Articles

On leastp-th power methods in multiple regressions and location estimations

The solution of the matricial systemZ=HX+V is derived. The observations are given by the matrixZ(n×m) and arem-components vectors. The parameter matrix isH(n×g), whileX(g×m) is the unknown andV(n×m) the residual matrix. The solutionX is estimated in order to minimize anLp-norm ofV defined with the help of anLr-norm on theV rows. The minimization algorithm is a steepest descent method with a specially chosen step size to “avoid” the singularities. The general problem shrinks in simple regression or location problems whilem org is reduced to the unit.

Erratum—A Note on Sharpe's Algorithm for Minimizing the Sum of Absolute Deviations in a Simple Regression Problem

Erratum to the authors' paper “A Note on Sharpe's Algorithm for Minimizing the Sum of Absolute Deviations in a Simple Regression Problem”, Vol. 19, No. 2 (October 1972), pp. 222–225.

A Note on Sharpe's Algorithm for Minimizing the Sum of Absolute Deviations in a Simple Regression Problem

A recent article in this Journal provides an efficient computational scheme for obtaining a regression line that minimizes the sum of absolute deviations of a set of two-dimensional points. The present note interprets that procedure as the solution of the parametric dual to the linear programming formulation of the problem. An alternate and about equally efficient procedure for solving the same problem is also provided.

The Interpolation of Time Series by Related Series

Abstract The construction of most comprehensive economic time series involves the estimation of some components for some dates by interpolation between values (“benchmarks”) for earlier and later dates. This is often done by using a related series known for all relevant dates. In practice, the bulk of such interpolation uses only a single related series, its values for only three dates—the interpolation date, one prior date, and one later date—and the values of the given series for only the prior and later dates. For this special case, some current methods are defective on formal grounds and all may frequently yield less accurate estimates than straight-line or other mathematical interpolation. Converting the problem into a simple bivariate regression problem suggests a generalization of current methods that takes account of the correlation between the movements of the given series and the related series.

The Interpolation of Time Series by Related Series

Abstract The construction of most comprehensive economic time series involves the estimation of some components for some dates by interpolation between values (“benchmarks”) for earlier and later dates. This is often done by using a related series known for all relevant dates. In practice, the bulk of such interpolation uses only a single related series, its values for only three dates—the interpolation date, one prior date, and one later date—and the values of the given series for only the prior and later dates. For this special case, some current methods are defective on formal grounds and all may frequently yield less accurate estimates than straight-line or other mathematical interpolation. Converting the problem into a simple bivariate regression problem suggests a generalization of current methods that takes account of the correlation between the movements of the given series and the related series.

Some Nonparametric Median Procedures

Under the nonparametric approach, various methods have been suggested to avoid the assumption of normality and homoscedasticity implicit in the analysis of variance. For the one-way classification, i.e., to decide whether $c$ samples come from the same population, Kruskal and Wallis [9] have proposed the $H$-test based on ranks; Mood and Brown [10] have proposed the $M$-test, utilizing the numbers of observations above the median of the combined sample; while the present author [2] has offered the $V$-test based on the number of $c$-plets that can be formed by choosing one observation from each sample such that the observation from the $k$th sample is the least, $k = 1, 2, \cdots, c$. For the two-way classification, Friedman [8] has made use of ranks. His statistic, to test the hypothesis that the rankings by $m$ "observers" of $n$ "objects" are independent, essentially offers a test for the two-way classification with one observation per cell. Durbin [7] has given a generalization for the balanced incomplete block design. Benard and Van Elteren [1] have generalized it still further. Mood and Brown [6, 10] also have considered the two-way classification with one observation per cell or the same number of observations per cell. In the first part of the present paper, their test has been extended to cover incomplete block situations. Mood and Brown [6, 10] have also considered some simple regression problems. In the present paper their methods are extended to discuss some additional regression problems. Next some bivariate analysis of variance problems are considered. The "step-down procedure" [11, 12] is used to reduce the problem to the univariate case with the other variate as a concomitant variate. The regression methods developed earlier are used here in these bivariate problems. The method seems to be perfectly general and could be extended to three or more variates, that is, to the multivariate situation. Most of the tests are offered on heuristic considerations. They are expected to be good for large samples.

Tests of significance in the simple regression problem

The orthogonal transformation used in a previous paper(1) to derive the distribution ofx2has its counterpart in a double orthogonal transformation which can be applied to a variableydependent on another variablexin order to deduce the known tests of significance in the simple regression problem. The general theorem is due to R. A. Fisher (2), and the particular transformation now used is a special case of the problem considered by Vajda (3) in the multivariate case. Attention is also directed to papers by Bartlett (4) and Elfving (5). In the present problem the use of two successive transformations is of interest in showing what happens, and in providing a relatively elementary proof of the known results.

An extension of the Doolittle method to simple regression problems.

An extension of the Doolittle method to simple regression problems.