Asymptotic Linearity of Wilcoxon Signed-Rank Statistics

Abstract

As a "robust" alternative to the least squares estimates for a regression parameter, Koul (1969) proposed new estimates based on signed-rank statistics. To find out their asymptotic distribution Koul proved that under quite general assumptions, the signed-rank statistics of Wilcoxon type are asymptotically linear in the sense that they are uniformly approximable by linear forms in the regression parameter. More general results have been obtained by Van Eeden (1972) in a paper which is an analog to Jureckova's paper (1969) dealing with linear rank statistics. In 1972 the author proved that the statistic used to define the Hodges-Lehmann estimate for a location parameter is asymptotically linear in a stronger sense, the result being to Koul's theorem what the central limit theorem is to the weak law of large numbers. For the general linear regression model with one parameter the signed-rank statistics are proved to be linear in a strong sense, that is, the differences between the statistics and the linear forms mentioned above, properly normalized, converge weakly to linear processes. Results in this direction for linear rank statistics have been obtained by Jureckova (1973). As an application of the theorems presented here, one can construct new estimates for the squared $L_2$-norm of the underlying density, and this in much the same way as in Antille (1974). It is also possible to get more information about the asymptotic behavior of the linearized versions, proposed by Kraft and Van Eeden (1972).