Combining Independent Estimators and Estimation in Linear Regression with Unequal Variances

A recent method, called MINQUE, is applied to two important problems: (1) combining k independent estimators gi (i = 1, * *, k) of a parameter A, where Yi is the arithmetic mean of ni (>1) observations normally and independently distributed with mean ,u and variance a,; (2) estimating the parameters a and f in the regression model (with replicates) Yij = a + ,xii + eii, j = 1, * , ni, i-1, k * *X, where the xi are known constants and the eii are normally and independently distributed with mean 0 and varianceo2. Two simple modifications of MINQUE which guarantee positive estimates of oa are given. We empirically investigate the relative efficiency of MINQUE over weighted least squares (WLS) estimators (using the sample variance s2 to estimate o-2) and maximum likelihood (ML) estimators. A major conclusion is that MINQUE (with modifications) lead to large gains in efficiency over WLS estimators when ni = m is small and k is relatively large. Another important result is that MINQUE may not lead to substantial gains in efficiency when m is >8, especially for small k. A first order approximation for estimator of variance performed better than that for WLS estimator.