Erich Lehmann on rank methods

Abstract

Though statistical methods based on ranks, such as the sign test, have a history going back for centuries, it was the introduction of Wilcoxon's tests in Wilcoxon (1945) that started an avalanche of research on this subject. It is worth noting that Wilcoxon needed only a little over three pages introducing and discussing both his two-sample test and his one-sample test for symmetry, to set off a research effort that would fill thousands of journal pages. Almost from the beginning Erich Lehmann played a major role in the development of this subject. Eight of his papers will be discussed here, of which two are joint with J. L. Hodges, Jr. and one with C. Stein. The full content of these papers will not be described here. Instead, the focus will be on the main contributions of each paper to our present knowledge of rank tests. It all adds up to what is more or less a history of the subject. For ease of presentation attention will be focused on the two-sample problem. When needed, appropriate reference will be made to other models that are dealt with in a particular paper. Let us introduce the necessary notation. The two samples are denoted by X 1, X2, ... , Xm and Yt, Y2, ... , Yn and for the combined sample of size N = m + n, we write Zt, ... , ZN = X1, . .. , Xm, Y1, ... , Yn. We assume that Z1, ... , ZN are independent random variables, that X1, X2, ... , Xm are identically distributed (i.d.) with common distribution function F, and that Y1, Yz, ... , Yn are i.d. with distribution function G. We suppose that F and G are continuous so that there are no equal values among Z 1, • • • , Z N . Let Zcll < Z(N) denote the ordered sequence of Z 1, • • • , ZN, and R 1 the rank of Y 1 in this sequence, thus Y 1 = Z(Rj)· Wilcoxon's