A QUICK ESTIMATE OF THE REGRESSION COEFFICIENT

1. In this note we investigate some properties of a 'quick' estimator, b', of the regression coefficient /? of a variable y on a variable x. This estimate has the advantage over the least squares estimate that (a) it is applicable to certain types of censored data, and (b) it provides a consistent estimator (under certain restrictions) of the slope parameter when y and x are structurally, rather than regressively, related (in the sense of Kendall (195 1-2) and Neyman & Scott (1951). It has been chiefly considered in relation to problems of this latter kind (Wald, 1940; Banerjee & Nair, 1942; Bartlett, 1949; Hidimoto, 1956), although it originated in regard to the situation where the x's were arbitrary constants (Bose, 1938; Nair & Shrivastava, 1942). We shall consider its behaviour when x and y are samples from a bivariate population. We show that it has an efficiency of 75-80 % when the population is bivariate normal by means of a conditional expectation technique which enables the results of David & Johnson (1954) to be applied.* We further examine its distribution from the point of view of its providing a quick test of the hypothesis /? = 0. 1 1. We have a set of n independent observations of (x, y) where