Robustness of Some Nonparametric Procedures in Linear Models

Abstract

For the random variables $X_{ij}(i = 1, \cdots, N; j = 1, \cdots, r)$ consider the linear model \begin{equation*}\tag{1.1} X_{ij} = \mu + \beta_i + \tau_j + Y_{ij} (\sum \beta_i = 0, \sum\tau_j = 0),\end{equation*} where the $\tau$'s are treatment effects, the $\beta$'s are nuisance parameters (block effects), and the $Y_{ij}$'s are error components. Nonparametric procedures for estimating and testing contrasts in the $\tau$'s, based on the Wilcoxon signed rank statistics, are due to Lehmann (1964), Hollander (1967) and Doksum (1967), among others. These rest on the assumption that the $Y_{ij}$'s are independent with a common continuous distribution. Since these procedures are actually based on the paired differences $X^\ast_{ijk}$, defined by (2.1), they are unaffected by the addition of a random variable $V_i$ to $\beta_i$ (or to $Y_{ij}$) for $i = 1, \cdots, N$. The object of the present investigation is to show that these procedures are valid even if $Y_{i1}, \cdots, Y_{ir}$ are interchangeable random variables, for each $i( = 1, \cdots, N)$. It may be noted that if in (1.1) the superimposed random variable $V_i$ is absorbed in $Y_{ij}$, then of course $Y_{i1}, \cdots, Y_{ir}$ are interchangeable, but the interchangeability of $Y_{i1},\cdots, Y_{ir}$ does not necessarily imply that $Y_{ij} = W_{ij} + V_i$, where $W_{ij}$'s are independent and identically distributed random variables (iidrv). In fact, in `mixed model' experiments, interchangeability of $Y_{i1}, \cdots, Y_{ir}$ (of quite arbitrary nature) may arise when there is no block versus treatment interaction [cf. Koch and Sen (1968) for details]. It is also shown that the procedures mentioned above are robust against possible heterogeneity of the distributions of the error vectors $\mathbf{Y}_i = (Y_{i1}, \cdots, Y_{ir}), i = 1, \cdots, N$. This situation may arise when the block effects are not additive or the errors are heteroscedastic. Thus, in this paper the independence of the errors is replaced by within block symmetric dependence, while the additivity of the block effects and homoscedasticity of the errors are relaxed.