Robustness of the Wilcoxon Estimate of Location Against a Certain Dependence

Abstract

Let $Z_1, Z_2, \cdots, Z_N$ be $N$ random variables with common distribution $P(Z_i \leqq u) = F(u - \theta)$ where $F \varepsilon \mathscr{F}_{cs0}$ which throughout this paper means the class of all continuous distributions symmetric about zero. The distributions considered will furthermore satisfy the regularity conditions of Lemma 3a in [3]. $\theta$ is an unknown constant to be estimated. In the case where $Z_1, \cdots, Z_N$ are independent, the following estimate of $\theta$ has been recently investigated by J. L. Hodges Jr. and E. L. Lehmann [4]. \begin{equation*}\tag{0.1}\theta^{\ast} = \operatorname{med}_{r\leqq s} \{(Z_r + Z_s)/2\},\end{equation*} .e. the median of the $N + \binom{N}{2}$ averages $(Z_r + Z_s)/2$. The asymptotic efficiency of $\theta^{\ast}$ relative to the classical estimate \begin{equation*}\tag{0.2}\hat{\theta} = \sum^N_{i=1} Z_i/N\end{equation*} in the sense of reciprocal ratio of asymptotic variances has been determined in [4] and shown to be $12\sigma^2_z\lbrack\int f{}^2(z) dz\rbrack^2$ where $f$ is the density corresponding to $F$ and $\sigma_z^2$ denotes the variance of the $Z$'s. It follows directly from Theorem 2.2 of [6] that $\theta^{\ast}$, in case of independent but not necessarily symmetrically distributed observations $Z_j$, is a consistent estimate of the pseudomedian of $F$ ([6], p. 178), which in general may be different from the median $\theta$. In this paper we shall consider a situation where only few $(c)$ observations can be collected per day and where the experiments have to be conducted over several $(n)$ days to yield the necessary number of observations. During this period the experimental conditions may easily change, whereby the standard assumption of "independent and identically distributed" observations is violated. The data occur naturally grouped in $n$ blocks, $c$ observations per block, and the possible change of conditions is introduced as a (nuisance) random block effect. We shall study the behavior of the two estimates $\theta^{\ast}$ and $\hat\theta$ under such conditions to find out how robust they are against this kind of dependence. In particular we shall study their asymptotic behavior as $n \rightarrow \infty$ with $c$ fixed, and shall derive a general expression for the asymptotic efficiency of $\theta\ast$ relative to $\hat{\theta}$. The efficiency is finally computed for normal and gross error models.