Some New Distribution-Free Statistics

Abstract

For the two-sample problem, Wilcoxon [21], Fisher and Yates [6], Terry [19], Hoeffding [10], Hodges and Lehmann [8], Savage [17], Chernoff and Savage [3], Lehmann [13], Capon [2] and others have considered rank-sum statistics equivalent to $S_N(H) = m^{-1} \sum E(Z(R(X_i)) \mid H) - n^{-1} \sum E(Z(R(Y_i)) \mid H)$, where $E(Z(j) \mid H)$ is the expectation of the $j$th order statistic of a sample of size $N = m + n$ from a population with cpf (cumulative probability function) $H$ and $R(X_i) \lbrack R(Y_j)\rbrack$ is the rank of $X_i\lbrack Y_j\rbrack$ in the combined sample of $X$'s and $Y$'s. In order to perform tests based on these statistics one needs special tables of expected values as well as tables of the hypothesis distribution. Further, in general, exact desired significance levels can only be achieved through randomization. The object of this note is to introduce rank-sum statistics in which one randomizes initially and circumvents the necessity of two special tables. These new randomized statistics, which are formed by deleting the expectation signs "$E$" in $S_N(H)$, generally satisfy the same asymptotic goodness criteria as their nonrandomized counterparts. Moreover, they have the added advantage that for appropriate choices of the parameters they have null hypothesis distributions which are continuous, known and tabulated (e.g., normal, $\chi^2$, etc.) In particular, one of the new two-sample statistics has an exact normal distribution and is asymptotically uniformly more efficient than the $t$-test for translation alternatives. This idea is extended to obtain randomized rank-sum statistics for the independence, randomness, $k$-sample and two-factor problems analogous to the statistics of Friedman [7], Puri [15], Stuart [18] and others. As in the references listed above, prime interest will be in those cases for which $H$ is normal, uniform or exponential. However, the methodology is equally applicable to other continuous cpf's $H$.