By Klaus Hinkelmann, Oscar Kempthorne
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Extra resources for Design and Analysis of Experiments, Advanced Experimental Design (Wiley Series in Probability and Statistics) (Volume 2)
It is therefore not surprising that so far no complete answer has been given. The most general result for the Yates estimator (and a somewhat larger class of estimators) is that of Shah (1964) based upon some general results by Roy and Shah (1962) [see also Bhattacharya (1998) for proper designs]. It is shown there that the combined estimator for any treatment contrast in any (proper) incomplete block design has variance smaller than that of the corresponding intrablock estimator if ρ does not exceed 2, or, equivalently, if σβ2 ≤ σe2 /k.
Since this method was proposed first by Yates (1940a) we shall refer to it as the Yates procedure or to the estimators as the Yates estimators. 1 Yates Procedure One way to estimate w and wj is to first estimate σe2 and σβ2 and then use these estimators to estimate w and wj . If the estimators are denoted by σe2 and σβ2 , respectively, then we estimate w and wj as w= 1 σe2 wj = σe2 1 + kj σβ2 (j = 1, 2, . . 2. 80) to obtain w and wj (j = 1, 2, . . , b). 61). 64). 3. 61) are replaced by their estimators will have an effect on the properties of the estimators for treatment effects.
Since ψ = y − , we have var Xβ ψ = var X β y = Xβ Xβ 2 σβ2 + X β Xβ σe2 = σe2 K + γ K 2 Hence ψ P V −1 P ψ is equal to Q2 ≡ [B − E(B)] K + γ K 2 with B = Xβ y being the vector of block totals. 75) We thus summarize: To obtain the intrablock estimator, we minimize Q1 ; to obtain the interblock estimator, we minimize Q2 ; and to obtain the combined estimator, we minimize Q1 + Q2 . 2 Case of Proper, Equireplicate Designs We now consider the case with R = rI and K = kI . 76) shows again two things: 1.