Statistics & Probability Letters 58 (2002) 53–59 On characterization of two-sample U -statistics E. Schechtman a ; ∗ , G. Schechtman b a Department of Industrial Engineering and Management, Ben Gurion University of the Negev, Beer Sheva, Israel b Department of Mathematics, The Weizmann Institute, Rehovot, Israel Received November 2001 Abstract A veriable condition for a symmetric statistic to be a two-sample U -statistic is given. As an illustration, we characterize which linear rank statistics with two-sample regression constants are U -statistics. We also show that invariance under jackkning characterizes a two-sample U -statistic. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Jackknife; Kernel; Linear rank statistic 1. Introduction Let(X 1 ;:::;X n ; Y 1 ;:::;Y m ) be two independent random samples from distributions with c.d.f’s F (x) and G(y), respectively. Let be a parameter and suppose that (r;t ) are the smallest sample sizes for which there exists a function h, symmetric in its rX ’s and tY ’s, such that E(h(X 1 ;:::;X r ; Y 1 ;:::;Y t ))=,forevery F;G in some family of distributions. Then, the two-sample U -statistic with kernel h, and of degree (r;t ), is, for n ¿ r; m ¿ t , U n;m = U (X 1 ;:::;X n ; Y 1 ;:::;Y m ) = n r -1 m t -1 (n;r) (m;t ) h(X i1 ;:::;X ir ; Y j1 ;:::;Y jt ); where ∑ (n;r) ( ∑ (m;t ) ) denotes summation over all distinct subsets i 1 ;:::;i r of the integers 1;:::;n ( j 1 ;:::;j t of the integers 1;:::;m). U -statistics were rst studied by Hoeding (1948), and were further investigated by many authors. (See, for example, Randles and Wolfe, 1979; Sering, 1980; Lee, 1990). Once a statistic is known * Corresponding author. Fax: +972-8-647-2958. E-mail addresses: ednas@bgumail.bgu.ac.il (E. Schechtman), gideon@wisdom.weizmann.ac.il (G. Schechtman). 0167-7152/02/$-see front matter c 2002 Elsevier Science B.V. All rights reserved. PII:S0167-7152(02)00110-4