How do bootstrap and permutation tests work? (English) Zbl 1028.62027

Janssen, Arnold; Pauls, Thorsten

Ann. Stat. 31, No. 3, 768-806 (2003).

Summary: Resampling methods are frequently used in practice to adjust critical values of nonparametric tests. In the present paper a comprehensive and unified approach for the conditional and unconditional analysis of linear resampling statistics is presented. Under fairly mild assumptions we prove tightness and an asymptotic series representation for their weak accumulation points. From this series it becomes clear which part of the resampling statistic is responsible for asymptotic normality. The results lead to a discussion of the asymptotic correctness of resampling methods as well as their applications in testing hypotheses. They are conditionally correct iff a central limit theorem holds for the original test statistic. We prove unconditional correctness iff the central limit theorem holds or when symmetric random variables are resampled by a scheme of asymptotically random signs.
Special cases are the \(m(n)\) out of \(k(n)\) bootstrap, the weighted bootstrap, the wild bootstrap and all kinds of permutation statistics. The program is carried out for convergent partial sums of rowwise independent infinitesimal triangular arrays in detail. These results are used to compare power functions of conditional resampling tests and their unconditional counterparts. The proof uses the method of random scores for permutation type statistics.

MSC:

62G09	Nonparametric statistical resampling methods
62G10	Nonparametric hypothesis testing
60F05	Central limit and other weak theorems
62G20	Asymptotic properties of nonparametric inference

Keywords:

sample mean; bootstrap; weighted bootstrap; wild bootstrap; permutation statistics; resampling tests; conditional tests; random scores; rank statistics; infinitely divisible laws; two-sample test; exchangeable variables

References:

[1]	ARAUJO, A. and GINÉ, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York. · Zbl 0457.60001
[2]	ARCONES, M. and GINÉ, E. (1989). The bootstrap of the mean with arbitrary bootstrap sample size. Ann. Inst. H. Poincaré Probab. Statist. 25 457-481. · Zbl 0712.62015
[3]	ARCONES, M. and GINÉ, E. (1991). Additions and corrections to ”The bootstrap of the mean with arbitrary bootstrap sample size.” Ann. Inst. H. Poincaré Probab. Statist. 27 583-595. · Zbl 0747.62019
[4]	ATHREy A, K. B. (1987). Bootstrap of the mean in the infinite variance case. Ann. Statist. 15 724-731. · Zbl 0628.62042 · doi:10.1214/aos/1176350371
[5]	BARBE, P. and BERTAIL, P. (1995). The Weighted Bootstrap. Lecture Notes in Statist. 98. Springer, New York. · Zbl 0826.62030
[6]	BERAN, R. (1988). Prepivoting test statistics: A bootstrap view of asy mptotic refinements. J. Amer. Statist. Assoc. 83 687-697. JSTOR: · Zbl 0662.62024 · doi:10.2307/2289292
[7]	BICKEL, P. J., GÖTZE, F. and VAN ZWET, W. R. (1997). Resampling fewer than n observations: Gains, losses, and remedies for losses. Statist. Sinica 7 1-31. · Zbl 0927.62043
[8]	BILLINGSLEY, P. (1971). Weak Convergence of Measures: Applications in Probability. SIAM, Philadelphia. · Zbl 0271.60009
[9]	CHERNOFF, H. and TEICHER, H. (1958). A central limit theorem for sums of interchangeable random variables. Ann. Math. Statist. 29 118-130. · Zbl 0085.35102 · doi:10.1214/aoms/1177706709
[10]	CSÖRG O, M., CSÖRG O, S., HORVÁTH, L. and MASON, D. M. (1986). Normal and stable convergence of integral functions of the emirical distribution function. Ann. Probab. 14 86-118. · Zbl 0589.60030 · doi:10.1214/aop/1176992618
[11]	CSÖRG O, S., HÄUSLER, E. and MASON, D. M. (1988). A probabilistic approach to the asy mptotic distribution of sums of independent, identically distributed random variables. Adv. in Appl. Math. 9 259-333. · Zbl 0657.60029 · doi:10.1016/0196-8858(88)90016-4
[12]	CSÖRG O, S. and MASON, D. M. (1989). Bootstrapping empirical functions. Ann. Statist. 17 1447- 1471. · Zbl 0701.62057
[13]	CUESTA-ALBERTOS, J. A. and MATRÁN, C. (1998). The asy mptotic distribution of the bootstrap sample mean of an infinitesimal array. Ann. Inst. H. Poincaré Probab. Statist. 34 23-48. · Zbl 0907.62018 · doi:10.1016/S0246-0203(98)80012-5
[14]	DEHEUVELS, P., MASON, D. M. and SHORACK, G. R. (1993). Some results on the influence of extremes on the bootstrap. Ann. Inst. H. Poincaré Probab. Statist. 29 83-103. · Zbl 0774.62042
[15]	DEL BARRIO, E. and MATRÁN, C. (2000). The weighted bootstrap mean for heavy-tailed distributions. J. Theoret. Probab. 13 547-569. · Zbl 1163.62320 · doi:10.1023/A:1007885222438
[16]	DEL BARRIO, E., MATRÁN, C. and CUESTA-ALBERTOS, J. A. (1999). Neccessary conditions for the bootstrap of the mean of a triangular array. Ann. Inst. H. Poincaré Probab. Statist. 35 371-386. · Zbl 0930.62047
[17]	DUDLEY, R. M. (1989). Real Analy sis and Probability. Wadsworth, Pacific Grove, CA.
[18]	EINMAHL, U. and MASON, D. M. (1992). Approximations to permutation and exchangeable processes. J. Theoret. Probab. 5 101-126. · Zbl 0791.60037 · doi:10.1007/BF01046780
[19]	GINÉ, E. and ZINN, J. (1989). Necessary conditions for the bootstrap of the mean. Ann. Statist. 17 684-691. · Zbl 0672.62026 · doi:10.1214/aos/1176347134
[20]	GNEDENKO, B. V. and KOLMOGOROV, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA. · Zbl 0056.36001
[21]	HÁJEK, J. (1961). Some extensions of the Wald-Wolfowitz-Noether theorem. Ann. Math. Statist. 32 506-523. · Zbl 0107.13404
[22]	HÁJEK, J. (1968). Asy mptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39 325-346. · Zbl 0187.16401 · doi:10.1214/aoms/1177698394
[23]	HÁJEK, J., SIDÁK, Z. and SEN, P. K. (1999). Theory of Rank Tests, 2nd ed. Academic Press, Orlando, FL.
[24]	HALL, P. (1990). Asy mptotic properties of the bootstrap for heavy-tailed distributions. Ann. Probab. 18 1342-1360. · Zbl 0714.62035 · doi:10.1214/aop/1176990748
[25]	HALL, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York. · Zbl 0744.62026
[26]	HARTMAN, P. and WINTNER, A. (1942). On the infinitesimal generators of integral convolutions. Amer. J. Math. 64 273-298. JSTOR: · Zbl 0063.01951 · doi:10.2307/2371683
[27]	JANSSEN, A. (1989). Local asy mptotic normality for randomly censored models with applications to rank tests. Statist. Neerlandica 43 109-125. · Zbl 0715.62094 · doi:10.1111/j.1467-9574.1989.tb01252.x
[28]	JANSSEN, A. (1991). Conditional rank tests for randomly censored data. Ann. Statist. 19 1434-1456. · Zbl 0729.62043 · doi:10.1214/aos/1176348255
[29]	JANSSEN, A. (1994). Sums of independent triangular array s and extreme order statistics. Ann. Probab. 22 1766-1793. · Zbl 0836.60012 · doi:10.1214/aop/1176988482
[30]	JANSSEN, A. (1997). Studentized permutation tests for non-i.i.d. hy potheses and the generalized Behrens-Fisher problem. Statist. Probab. Lett. 36 9-21. · Zbl 1064.62526 · doi:10.1016/S0167-7152(97)00043-6
[31]	JANSSEN, A. (1999). Nonparametric sy mmetry tests for statistical functionals. Math. Methods Statist. 8 320-343. · Zbl 1103.62343
[32]	JANSSEN, A. (2000). Invariance principles for sums of extreme sequential order statistics attracted to Lévy processes. Stochastic Process. Appl. 85 255-277. · Zbl 0997.60031 · doi:10.1016/S0304-4149(99)00078-2
[33]	JANSSEN, A. and MASON, D. M. (1990). Non-standard Rank Tests. Lecture Notes in Statist. 65. Springer, New York. · Zbl 0711.62038
[34]	KINATEDER, J. (1992). An invariance principle applicable to the bootstrap. In Exploring the Limits of Bootstrap (R. LePage and L. Billard, eds.) 157-181. Wiley, New York. · Zbl 0846.60033
[35]	KNIGHT, K. (1989). On the bootstrap of the sample mean in the infinite variance case. Ann. Statist. 17 1168-1175. · Zbl 0687.62017 · doi:10.1214/aos/1176347262
[36]	LEPAGE, R., WOODROOFE, M. and ZINN, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632. · Zbl 0465.60031 · doi:10.1214/aop/1176994367
[37]	MAMMEN, E. (1992a). When Does Bootstrap Work? Asy mptotic Results and Simulations. Lecture Notes in Statist. 77. Springer, New York.
[38]	MAMMEN, E. (1992b). Bootstrap, wild bootstrap, and asy mptotic normality. Probab. Theory Related Fields 93 439-455. · Zbl 0766.62021 · doi:10.1007/BF01192716
[39]	MASON, D. M. and NEWTON, M. A. (1992). A rank statistics approach to the consistency of a general bootstrap. Ann. Statist. 20 1611-1624. · Zbl 0777.62045 · doi:10.1214/aos/1176348787
[40]	MASON, D. M. and SHAO, Q.-M. (2001). Bootstrapping the Student t-statistic. Ann. Probab. 29 1435-1450. · Zbl 1010.62026 · doi:10.1214/aop/1015345757
[41]	NEUHAUS, G. (1988). Asy mptotically optimal rank tests for the two-sample problem with randomly censored data. Comm. Statist. Theory Methods 17 2037-2058. · Zbl 0658.62063 · doi:10.1080/03610928808829731
[42]	NEUHAUS, G. (1993). Conditional rank tests for the two-sample problem under random censorship. Ann. Statist. 21 1760-1779. · Zbl 0793.62027 · doi:10.1214/aos/1176349396
[43]	PETROV, V. V. (1995). Limit Theorems of Probability Theory. Clarendon Press, Oxford. · Zbl 0826.60001
[44]	RAIKOV, D. A. (1938). On a connection between the central limit law of the theory of probability and the law of large numbers. Bull. Acad. Sci. URSS Ser. Math. 2 323-338. · Zbl 0019.22401
[45]	ROMANO, J. P. (1989). Bootstrap and randomization tests of some nonparametric hy potheses. Ann. Statist. 17 141-159. · Zbl 0688.62031 · doi:10.1214/aos/1176347007
[46]	SHORACK, G. R. (1996). Linear rank statistics, finite sampling, permutation tests and winsorizing. Ann. Statist. 24 1371-1385. · Zbl 0862.62017 · doi:10.1214/aos/1032526974
[47]	SHORACK, G. R. (2000). Probability for Statisticians. Springer, New York. · Zbl 0951.62005 · doi:10.1007/b98901
[48]	STRASSER, H. and WEBER, C. (1999). The asy mptotic theory of permutation statistics. Math. Methods Statist. 8 220-250. · Zbl 1103.62346
[49]	TUCKER, H. G. (1975). The supports of infinitely divisible distribution functions. Proc. Amer. Math. Soc. 49 436-440. JSTOR: · Zbl 0308.60014 · doi:10.2307/2040661
[50]	VAN DER VAART, A. W. and WELLNER, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York. · Zbl 0862.60002
[51]	WELLNER, J. (2001). Some converse limit theorems for exchangeable bootstraps. In State of the Art in Probability and Statistics. Festschrift for Willem R. van Zwet (M. de Gunst et al., eds.) 593-606. IMS, Beachwood, OA. · Zbl 1380.62185 · doi:10.1214/lnms/1215090092
[52]	WITTING, H. and NÖLLE, G. (1970). Angewandte Mathematische Statistik. Optimale finite und asy mptotische Verfahren. Teubner, Stuttgart. · Zbl 0212.20901
[53]	ZOLOTAREV, V. M. and KRUGLOV, V. M. (1975). The structure of infinitely divisible distributions on a bicompact Abelian group. Theory Probab. Appl. 20 698-709. · Zbl 0358.60027 · doi:10.1137/1120079