Some Robust Selection Procedures

Abstract

Let $X_{it} (t = 1,\cdots, n; i = 1,\cdots, k)$ be independent observations from $k$ populations with respective distribution functions $F(x - \theta_i)$, where the translation parameters $\theta_i$ are unknown. Consider the problem of selecting one population, the objective being to select the population with largest translation parameter. Procedures based on the joint ranking of all $nk$ observations have been considered by Lehmann [5], Bartlett and Govindarajulu [1], and Puri and Puri [9]. Robust procedures for related problems have been considered by Sobel [11] and McDonald and Gupta [7], among others. Define the $i$th population to be good if $\theta_i > \theta_{\max} - \Delta$ where $\theta_{\max} = \max \{\theta_1,\cdots, \theta_k\}$ and where $\Delta$ is a specified positive constant. The asymptotic relative efficiency (A.R.E.) of two procedures is then the limiting ratio of the sample sizes required to achieve a preassigned minimum probability of selecting a good population. It was hoped that procedures based on ranks would be more robust in terms of A.R.E. than corresponding parametric procedures. However, it has recently been shown that the slippage configuration used to find the A.R.E. in references [5] and [1] was not least favorable for the selection of a good population (See reference [10].). Puri and Puri [9] avoided this difficulty by restricting consideration to parameter points $\mathbf{\theta}^{(n)} = (\theta_1^{(n)},\cdots, \theta^{(n)}_k)$ for which $\theta^{(n)}_{\max} - \theta^{(n)}_i = b_i/n^{\frac{1}{2}} + 0(1/n ^{\frac{1}{2}})$ for $i = 1,\cdots, k$, where the $b_i$ are nonnegative constants. In Section 2, selection procedures are defined which are based on two-sample estimates of shift. It is shown in Section 3 that if the underlying distribution $F(x)$ is absolutely continuous then the procedures defined in Section 2 will select a unique population. Conditions are given under which the slippage configuration is the least favorable parameter point for the selection of a good population. This result does not require restrictions on the set of translation parameters comprising the parameter space. The A.R.E. of these procedures is defined in Section 4. If we consider the procedure based on the Hodges-Lehmann estimates of shift corresponding to the two-sample $F_0$-scores test, it is shown that the A.R.E. of this procedure relative to the normal theory procedure of Bechhofer [2] is simply the Pitman efficiency of the two-sample $F_0$-scores test relative to the $t$-test. Hence this approach yields efficiency results which are similar to those in references [5], [1], and [9]. However, the use of estimates in the definition of the selection procedure has the advantage of eliminating the difficulties concerning the least favorable parameter point.