On the Estimation of Heteroscedastic Variances

The method of minimum norm quadratic unbiased estimation (MINQUE) is applied to the estimation of the variances o-2 in the following problems: (1) Combining k independent estimators yi (i = 1, * , k) of a parameter g, where Ad is the arithmetic mean of ni (>1) observations Yi; (j = 1, * , no) normally and independently distributed with mean A and variance o-J; (2) estimating the parameters a and i3 in a linear regression model with replicates: yij = a + xix + ei1 where the et; are normally and independently distributed with mean 0 and variance o-J. The variance-covariance matrix of the MINQU estimators 6J is derived and, using the trace and determinant criteria, the gains in efficiency over the customary estimators s2 = (ni )-1 Es (yj gj)' are tabulated. A simple modification of 67, which is always positive and analytically tractable, performed better than the 6T, especially when k is small. The MINQU estimator of the variance of ordinary least squares (OLS) estimator of q (or a, Al) is obtained and the gains in efficiency over those estimators employing the s2 are evaluated. The MINQU estimators lead to substantial gains in efficiency when m is small and k is relatively large. Explicit expressions for the 62, which can be computed for any value of k, are given.