Statlstms & Probabdlty Letters 7 (1989) 179-185 North-Holland December 1988 TESTING FOR DISPERSIVE ORDERING Ibrahim A. AHMAD and Subhash C. KOCHAR * Dwtswn of Stattsttcs, Nothern Ilhnols Unwerslly, DeKalb, IL 60115, USA Recewed December 1987 Revased April 1988 dlsp Abstract An asymptotmally dxstnbutmn-free test is proposed and studied for testing the null hypothesis H 0 F = G versus dlsp H 1 F < G, that is G-l(fl)-G-l(a)>/F l(fl)_ F-l(a) for 0 ~< a < fl ~<1 The proposed test is based on an estimator of ]fZ(x) dz - fg2(x) dx, where f(g) is the probability density function corresponding to the distribution function F(G) The cases of two independent samples as well as that of pmred samples are discussed in detads A M S 1980 SubJect Classtficatton: Primary 62G05, 62E20, Secondary. 62G10 Keywords nonparametnc, asymptoUcally distribution-free, density estimate, kernel, window size 1. Introduction Let X and Y be two random variables having absolutely continuous distribution functions (d.f.'s) F and G, respectively, with F -a and G -1 as their left continuous inverses. Definition 1.1. G is said to be more dispersed than dlsp F, written F < G, if G-l(fl) - G-l(a) >_. F-l(fl) - F-l(a) forO~<a<fl~<l, (1.1) i.e., if any two quantiles of G are at least as far apart as the corresponding quantiles of F. The above definition is equix;alent to saying that G-1F(x)-x is nondecreasing in x. Doksum (1969) calls this ordering as tail ordering. Let f and g denote the probability density functions of F and G, respectively. The failure (hazard) rate of F is defined as rF(X )=f(x)/[1-F(x)], F(x)<I. * On leave from Panjab Umverslty, Chandlgarh, INDIA 0167-7152/88/$3 50 © 1988, Elsevier Science Pubhshers B V (North-Holland) Differentiating G-1F(x)- x, we find that (1.1) is equivalent to g[G-l(u)]<~f[F-a(u)] for 0~< u~< 1. (1.2) Equivalently, ro[G-a(u)] <~rF[F-t(u)] for0~<u~< 1. (1.3) The above several equivalent versions of disper- sive ordering have been discussed by many authors including Doksum (1969), Bickel and Lehmann (1979), Oja (1981), Lewis and Thompson (1981), and Shaked (1982). For positive random variables, which are mainly applicable in reliability theory, Deshpande and Kochar (1983), Bartoszewicz (1986), and Ahmad, Alzaid, Bartoszewicz, and Kochar (1986), have discussed relations between dispersive ordering, star ordering, superadditive ordering, and failure rate ordering. Note that if F(x)= H((x- 01)/,11) and G(x) dlsp =H((x-02)/'02), then F < G if and only if 1"/1 < "02. It is possible sometimes to compare two distri- butions F and G not necessarily belonging to the same location-scale family. Doksum (1969) has shown that for distributions symmetric about 0, if F is ordered with respect to G in the sehse of 179