Journal of Statistical Planning and Inference 139 (2009) 184 -- 202 Contents lists available at ScienceDirect Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi Resampling schemes with low resampling intensity and their applications in testing hypotheses Eustasio del Barrio a, ∗, 1 , Arnold Janssen b , Carlos Matr ´ an a, 1 a Universidad de Valladolid, 47005 Valladolid, Spain b Heinrich-Heine Universit ¨ at, D ¨ usseldorf, Germany ARTICLE INFO ABSTRACT Article history: Received 22 May 2007 Received in revised form 15 January 2008 Accepted 1 April 2008 Available online 09 April 2008 Keywords: Bootstrap Resampling Low intensity Exchangeable weights Two-sample permutation statistics Wild bootstrap Robust testing The paper explores statistical features of different resampling schemes under low resampling intensity. The original sample is considered in a very general framework of triangular arrays, without independence or equally distributed assumptions, although improvements under such conditions are also provided. We show that low resampling schemes have very interesting and flexible properties, providing new insights into the performance of widely used resampling methods, including subsampling, two-sample unbalanced permutation statistics or wild boot- strap. It is shown that, under regularity assumptions, resampling tests with critical values derived by the appertaining low resampling procedures are asymptotically valid and there is no loss of power compared with the power function of an ideal (but unfeasible) parametric family of tests. Moreover we show that in several contexts, including regression models, they may act as a filter for the normal part of a limit distribution, turning down the influence of outliers. © 2008 Elsevier B.V. All rights reserved. 1. Introduction In the present paper we introduce and study the concept of low resampling intensity for exchangeable resampling schemes. It is well known from the literature that sometimes Efron's ordinary bootstrap with resampling size m(n) equal to the sample size k(n) fails, while if m(n) = o(k(n)) it is consistent provided min(k(n), m(n)) →∞ as n →∞, see Athreya (1987) or Arcones and Gin ´ e (1989, 1991). In fact Mammen (1992a, b) showed (in an i.i.d. triangular-array setup) that, for linear statistics, consistency of the bootstrap with m(n) = k(n) is equivalent to asymptotic normality. See also Cuesta-Albertos and Matr ´ an (1998) and del Barrio et al. (1999) in relation with the general behavior of the bootstrap mean in this setup and Bickel et al. (1997) for a list of examples and further references regarding strategies to achieve bootstrap success. The bootstrap is part of more general resampling procedures given by exchangeable weights that, as it has progressively been made apparent, include well-known statistical methods as well as some new others suggested by the framework. The behavior of the weighted bootstrap mean has been considered for instance in Mason and Newton (1992), Praestgaard and Wellner (1993), Arenal-Guti ´ errez and Matr ´ an (1996), del Barrio and Matr ´ an (2000), Janssen and Pauls (2003) or Janssen (2005) (see the survey paper by Cs ¨ org ¨ o and Rosalsky, 2003 for other references). This general approach also covers two-sample permutation statistics, the wild bootstrap, or even subsampling (see e.g., Politis et al., 1999). Moreover, new proposals of resampling schemes designed for specific problems continuously appear in the literature. For example, Bose and Chatterjee (2005) consider generalized bootstrap ∗ Corresponding author. Tel.: +34 983423930; fax: +34 983423013. E-mail address: tasio@eio.uva.es (E. del Barrio). 1 These authors have been partially supported by the Spanish Ministerio de Educación y Ciencia and FEDER, Grant MTM2005-08519-C02-01,02 and the Consejería de Educación y Cultura de la Junta de Castilla y León, Grant PAPIJCL VA102A06. 0378-3758/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2008.04.002