A survey of two‐sample location‐scale problem, asymptotic relative efficiencies of some rank tests

Abstract

Summary Here the parametric as well as the non parametric approach to the two‐sample location‐scale problem are reviewed. Special attention is paid to the quadratic form of the linear rank statistics for the location and the scale suggested by Lepage for testing this problem. Several such quadratic forms are compared through ARE computations.

On the asymptotic properties of rank statistics for the two-sample location and scale problem

“…In the absence of Imp rank test for the problem (H, A), analogically to the para metric approach, the linear and quadratic form of the Imp rank test statistics 7\ and T 2 have been suggested in the literature; see Goria [4] for details. Clearly, the asymptotic power computations of these statistics are considerably simplified if the underlying pdf is symmetric about the origin.…”

“…Lepage [9] derives its distribution under the alternative A*, and shows further that Q x is asymptotically most powerful maximin test for testing H v against the alter native A*(3), where A*(S) is defined by the densities q v such that the pdff is symmetric about the origin and A^I^O) + All 22 (0) = 3. Duran et al [3] derive the expression for the asymptotic power efficiency of the statistics of the type Qi in the setting of Chernoff& Savage, while Goria [4] uses this expression and shows that it is better to use the statistic Q 1 among all statistics of this type, obtained through an arbitrary mixture of the rank statistics satisfying conditions (1.1).…”

“…Let (X l9 ... 9 X m ) and (X m+l9 ... 9 X N ) 9 N = m + n 9 be two independent random samples, and suppose that for some unknown value 9 = (9 l9 9 2 ) e R 2 the variable X x has the same absolutely continuous distribution F as 6 1 + X m+1 e~0 2 , F having con tinuously differentiable density f. Various authors namely Duran et al [3], Lepage [9], and Goria [4] have investigated the quadratic form of the linear rank statistics m $k = X a /<Nv^L)> k = 1 9 2 with regard to the problem of testing the hypothesis i=\ H : 9 = 0 against the location-scale alternative A : 9 =j = 0, where jR f is the rank of X t in the combined sample, S ± is the statistic for testing the difference in location, S 2 is the statistic for the scale problem. The former case corresponds to the alternative A x : 0 3 4= 0, 9 2 = 0, the latter one to A 2 : 9 X = 0, 9 2 + 0.…”