Yates type estimators of a common mean

Abstract

Letx, y, S, T andW be independent random variables such that,∼N(μασ ²),y∼N(μ,βη ²), S/σ²∼χ²(m), T/η²∼χ²(n) andW/(ασ ²+βη²)∼x²(q), where μ, σ², η² are unknown. For estimating μ, consider the estimator$\hat \mu = x + \left( {y - x} \right){{aS} \mathord{\left/ {\vphantom {{aS} {\left[ {S + cT + d\left\{ {\left( {y - x} \right)^2 + W} \right\}} \right],a,c,d > 0}}} \right. \kern-\nulldelimiterspace} {\left[ {S + cT + d\left\{ {\left( {y - x} \right)^2 + W} \right\}} \right],a,c,d > 0}}$. Note that the performance of$\hat \mu$ depends onτ=βη ²/ασ², which is unknown. Assumeq+n≧2 and leta ₀=(n+q−1)/(m+2), c^*=cα/β, d^*=dα. Two main results are:

• i) for all τ>0,$\hat \mu$ has a variance smaller than that ofx ifa≦2 min (1,c ^*a₀, d^*a₀);

• ii) for all τ≧τ₀, where τ₀>0 is arbitrary,$\hat \mu$ has a variance smaller than that ofx ifa≦2a ₀ min [c ^*τ₀/(1+τ₀),d^*].

We also obtain some necessary conditions for$\hat \mu$ to have a variance smaller than that ofx. It can be seen that with the exception of linked block designs for any design belonging to the class calledD ₁-class by Shah [16], Yates-Rao estimator for recovery of interblock information has the same form as that of$\hat \mu$. 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. For one of these two designs, we show that Yates' estimator is not uniformly better than the intra-block estimator.