Conditional-Normal Regression Models

Abstract

A relatively complete discussion is provided for the limiting distributions of certain sample correlation coefficients and sample correlation ratios. It is assumed for the random variables X and Y that the conditional distribution of Y , given X = x , is multivariate normal with a constant, but unknown, covariance matrix and that the distribution of X has finite fourth moments. The sample coefficients are then based on a random sample from the ( X, Y )-distribution. For the case of a univariate random variable X the limit laws are shown to depend on the X -distribution only through its coefficient of excess. In other cases they are determined from the coefficients of excess of univariate distributions closely related to the distribution of X.