This paper deals with two different problems. The first one deals with asymptotic normality of simple linear rank statistics based on random number of observations $X_i$, henceforth called random rank statistics, under the alternative where each $X_i$ has a different distribution $F_i$. The second problem deals with showing that the random rank statistics as a function of the regression parameter in the simple linear regression model is asymptotically uniformly linear (hence continuous) in that parameter. Obviously the two problems are different and could be solved in separate papers but for certain lemmas which are common to the solution of both of these problems. It is suggested not to try to apply the result of Section 3 to Section 4, unless mentioned explicitly. The results of Section 2 are the results which are common, to some extent, to the solution of these two problems. Pyke and Schorack  proved asymptotic normality of a class of two sample random rank statistics under two sample alternatives. Our theorem 3.1 could be thought of as a generalization of the result of  to more than two samples situation. Our score function $\varphi$ is in smaller class than that of . On the other hand our methods yield the asymptotic normality for random rank-sign statistics. This is also contained in Section 3. Section 2 proves a basic lemma about weak convergence of random weighted empirical cumulatives to a tied down continuous Gaussian process. In  and  asymptotic uniform linearity of rank statistics based on nonrandom number of observations was proved. In  conditions are very general on $\varphi$ and underlying distribution $F$ whereas conditions in  are quite stringent. But in  we do not need any artificial condition like (2.1) of  on underlying regression constants. However in both of these references, regression scores were assumed to be bounded. In Section 4 here we extend the results of  to random rank and random rank-sign statistics and to the case where regression scores need not be bounded. In Section 5 we show how the results of Section 4 can be used to construct a bounded length confidence interval for a regression parameter using rank sign statistics with asymptotic (as length $\rightarrow 0$) coverage probability achieved. Apart from applying Theorem 3.1 to the i.i.d. case, as is mentioned in the remark at the end of Section 5, it is hoped that Theorem 3.1 can be found applicable in some other interesting situations.
Hira Lal Koul. "Some Asymptotic Results on Random Rank Statistics." Ann. Math. Statist. 43 (3) 842 - 859, June, 1972. https://doi.org/10.1214/aoms/1177692550