Rank Tests with Estimated Scores and Their Application

Since the monograph of Hájek and Šidák (1967) it’s indisputable that linear rank tests form an attractive alternative to the classical tests based on normality assumptions, cf. Hollander and Wolfe (1973), Lehmann (1975), Conover (1980), and others. But meanwhile it has been demonstrated that the asymptotic optimality of linear rank tests for special types of alternatives is connected with rather low power for other types of alternatives. Therefore there have been some attempts to increase the power for certain classes of alternatives by adaptation, e.g. Randies and Hogg (1973) and others.

In the introduction and motivation of Chapter 1 we’ve discussed the merits and the shortcomings of linear rank tests by means of the two-sample problem. Similar arguments apply to other testing problems, too. Tests based on linear rank statistics S _N (b) have turned out to be optimal only for alternatives determined by the score function b. Since the alternative and thus the optimal 6 are unknown in reality, we suggest the following heuristic adaptation procedure: Instead of using a linear rank statistic S _N (b) with fixed score function b, we’ll estimate the optimal score function by some estimator b ̂ _N based on ranks, and use the nonlinear rank statistic S _N (b ̂ _N ) as the new test statistic.

In Chapter 1 we have motivated the need of new testing procedures in the case of two treatments differing in location. In Chapter 3 the corresponding mathematical foundation and the local asymptotic properties of the proposed testing statistics have been given. In this chapter we will discuss related problems. Therefore the basic parts of the motivation and the exposition are quite similar to those of Chapter 1 and Chapter 3.

As in Chapter 3 the basic problem is the comparison of a new treatment with a standard, but in contrast to Chapter 3 we don’t have two independent samples. Usually we have paired observations (Y _i , Z _i ), i = 1,..., n, where the Y-components are the measurements under the standard treatment whereas the Z -components are the measurements under the new treatment. It’s assumed that the design of the experiment allows the pairs (Y ₁, Z ₁),..., (Y _n , Z _n ) to be independent random variables with values in ℝ². It’s not realistic to assume the stochastic independence of the components Y _i and Z _i , since the Y _i -measurement and the Z _i -measurement are usually taken at matched pairs or even at the same experimental unit, e.g. left-right treatments or pre-post treatments, but we assume the differences Z ₁ - Y ₁,..., Z _n - Y _n to be i.i.d. random variables.

In this chapter we assume i.i.d. ℝ² -valued random variables X ₁,..., X _n with unknown (continuous) 2-dimensional distribution function F. Throughout the chapter we’ll use the following conventions and notations: If X _i is a ℝ² -valued random variable, then the components of X _i will be denoted by Y _i and Z _i , i.e. X _i = (Y _i , Z _i ).

The Appendix will contain some results and proofs which don’t fit into the main body of the text and where it’s difficult to find a suitable reference of the result used in the present setting.