Hodges-lehmann estimators in the case of grouped data - I
Let X' = (X'1,…,X'm) and Y' = (Y'1,…,Y'n) be two independent random samples from absolutely continuous distribution functions F' and G' respectively with G'(U) = F'(u-Δ) for all u. Let Xi and Yj be obtained from X'i and Y'j by rounding off to the nearest integer. Let h(X',Y') be any appropriate two-sample statistic. Let and , where N = m + n be the two-sample Hodges-Lehmann (H-L) estimators derived from h(X',Y') and h(X,Y), respectively. (In defining h(X,Y), ties are handled by the method of averaged scores.) We then have, (i) with probability 1. (ii) Prob as N → ∞. And, if Δ is an integer or zero, the stronger result Prob holds. Thus under the null hypothesis is asymptotically degenerate. The efficiency of relative to the usual parametric estimator, is, in a sense to be defined, the same as in the continuous case. If the measurements are very accurate (i.e. if they are rounded off to two or more decimals) , then the large-sample behaviour of is very nearly the same as . A more precise statement is given ...