Linear structural relationships with unknown error variances

Abstract

where (4, a + f) is the true value on the line and (?, () represents error of observation. With n independent observations (x1, y1), ..., (Xe, yn) of (x, y), we desire to estimate the parameter (o, ,B) of the straight line relationship under various assumptions on the observation errors and the true value structure. When the error variances are unknown, Geary (1942) proposed a method which makes use of the product cumulants to estimate the relationship. His method, however, has the drawback that it is difficult to decide which of the fourth-order product cumulants, namely K13, K22 and c31, to use (Kendall & Stuart, 1973, p. 412). It is even more serious that the estimates of the slope parameter ,B sometimes lie outside the regression limits for samples of moderate size. The geometric mean of the two regression limits has also been suggested as an estimator for /3, and yet it can easily be shown to be inconsistent. The linear structural relationship model usually refers to the case where the true value 4 is a random variable having a distribution with unknown parameters (Kendall & Stuart, 1973, Chapter 29). Here 4, s and ( are assumed to be independent random variables with s and ( normally distributed, having zero means and unknown variances a2 and 1A2, respectively. If 4 also has a normal distribution with unknown mean and variance, (x, y) has a bivariate normal distribution with six unknown parameters for which the sample means, variances and covariance are a set of sufficient statistics of dimension five only, yielding an unidentifiable situation for the parameters. On the other hand, if 4 is assumed to have a nonnormal distribution, the parameters are identifiable for the bivariate distribution of (x, y). Kiefer & Wolfowitz (1956) investigated various methods of obtaining consistent estimates of the slope /3 and yet, unfortunately, no method has been determined for calculating these estimates. For the particular case where 4 follows a uniform distribution over an interval [u T, p + f], they suggested that the method of maximum likelihood fails to estimate the parameters a, , a , i, ,u and T consistently. In their maximum likelihood approach, a global maximum is sought in optimizing the likelihood function. However, my investigations show that by locating a