Use of Inter-Block Information to Obtain Uniformly Better Estimators

Yates type estimators of a common mean

“…It is therefore, desirable that the combined estimator should be unbiased and have a variance which does not exceed that of the intra-block estimator for any possible values of the unknown variances of the individual estimators. From the work of Graybill and Weeks [8], Graybill and Seshadri [7] and Shah [16], Yates-Rao estimator is known to be unbiased but very little is known about its variance.…”

“…Of these, the earlier works of Graybill and Deal [6], Seshadri [14], [15], Shah [16] and Stein [19] all ignored some between block comparisons and moreover, their results did not cover all incomplete block designs. The recent works by Brown and Cohen [2] and Khatri and Shah [9] make use of all between block comparisons and while the scope of the former is limited to BIBD's only, the later applies to all incomplete block designs.…”

“…The recent works by Brown and Cohen [2] and Khatri and Shah [9] make use of all between block comparisons and while the scope of the former is limited to BIBD's only, the later applies to all incomplete block designs. The Yates-Rao procedure, which is still the most widely used, does utilize all between block comparisons and up till now, the only known design for which it fails to give uniform improvement over the intra-block estimator is the symmetric BIBD with four treatments and three replications (see Shah [16]). Simulation studies by El-shaarawi, Prentice and Shah [4] as well as the numerical comparisons by Khatri and Shah [10] indicate that for large designs as used in these studies, Yates-Rao estimator compares favorably with that of Khatri and Shah. In an earlier paper (Bhattacharya [1]), we considered a class of estimators which unified the two classes proposed in Brown and Cohen [2] and Khatri and Shah [9].…”

“…Hence, for such designs the above results can be used to examine if Yates estimator is good i.e., better than the intra-block estimator. Shah [16] resolved this question for linked block designs, which include the symmetrical BIBD's. Here, we consider asymmetrical BIBD's and show that Yates' estimator is good for all such designs listed in Fisher and Yates' table [5], with two exceptions.…”

Heteroskedasticity-robust estimation of means

“…A less general problem of estimating the common mean of two normal populations and the related problem of recovery of interblock information was initiated by Yates (1939Yates ( , 1940, and his work was extended by Nair (1944) and Rao (1947Rao ( , 1956, Related work in this area is due to Seshadri (1963,a,b,), Shah (1964), Stein (1966) and Khatri and Shah (1974). Seshadri (1963, b) develops a method of combining interblock and intrablock estimators into an estimator which is uniformly better in the variance sense than either single estimator alone.…”