Asymptotically Nonparametric Inference in some Linear Models with One Observation per Cell

Abstract

In [5] and [6] the author proposed point estimates, tests and confidence procedures for the parameters in a linear model, which have the same asymptotic efficiency (relative to the corresponding classical methods) as the Wilcoxon test has relative to the $t$-test. Here, "asymptotic" refers to the case that the numbers of observations per cell tend to infinity; in practice, they should presumably be at least equal to four. In the present paper, we shall consider experiments with only one observation per cell. It then of course becomes necessary to impose some restrictions on the parameters of the model. We shall suppose that an experiment, concerned with various factors at several levels, is replicated at different levels of a nuisance factor (i.e. in different "blocks") and that this nuisance factor does not interact with the factors of interest. Nonparametric procedures for this situation were considered earlier by Friedman (1937) and in [3], where, however, only tests were proposed for the hypothesis of no effect of the factors of interest. In the present paper we shall be concerned primarily with estimating arbitrary contrasts in the factors of interest, and also with the problem of testing such contrasts. Let us assume for the observations $X_{i\alpha}(i = 1, \cdots, c; \alpha = 1, \cdots, N)$ the model \begin{equation*}\tag{1.1}X_{i\alpha} = \nu + \xi_i + \mu_\alpha + U_{i\alpha} (\sum \xi_i = \sum \nu_\alpha = 0)\end{equation*} where the $\xi$'s are the parameters of interest, the $\mu$'s are the effects of the nuisance factor (or factors), and the $U$'s are independently distributed according to a common continuous distribution $F$. Except for the assumption of equal sample sizes, this agrees with the model of [5] and [6] when the $\mu$'s are assumed to be zero. We shall be interested in inference methods concerning the $\xi$'s, and their asymptotic properties as $N \rightarrow \infty$.