How do bootstrap and permutation tests work?

Annals of Statistics

Volumn 31, Issue 3, 2003, Pages 768-806

  Janssen, Arnold ^a   Pauls, Thorsten ^a  

^a HEINRICH HEINE UNIVERSITY   (Germany)

Author keywords

Bootstrap; Conditional tests; Exchangeable variables; Infinitely divisible laws; Permutation statistics; Random scores; Rank statistics; Resampling tests; Sample mean; Two sample test; Weighted bootstrap; Wild bootstrap

Indexed keywords

EID: 0041309838     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1056562462     Document Type: Article

References (53)

1. ARAUJO, A. and GINÉ, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.
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.
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.
4. ATHREYA, K. B. (1987). Bootstrap of the mean in the infinite variance case. Ann. Statist. 15 724-731.
5. BARBE, P. and BERTAIL, P. (1995). The Weighted Bootstrap. Lecture Notes in Statist. 98. Springer, New York.
6. BERAN, R. (1988). Prepivoting test statistics: A bootstrap view of asymptotic refinements. J. Amer. Statist. Assoc. 83 687-697.
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.
8. BILLINGSLEY, P. (1971). Weak Convergence of Measures: Applications in Probability. SIAM, Philadelphia.
9. CHERNOFF, H. and TEICHER, H. (1958). A central limit theorem for sums of interchangeable random variables. Ann. Math. Statist. 29 118-130.
10. CSÖRGO, M., CSÖRGO, 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.
11. CSÖRGO, S., HÄUSLER, E. and MASON, D. M. (1988). A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. Adv. in Appl. Math. 9 259-333.
12. CSÖRGO, S. and MASON, D. M. (1989). Bootstrapping empirical functions. Ann. Statist. 17 1447-1471.
13. CUESTA-ALBERTOS, J. A. and MATRÁN, C. (1998). The asymptotic distribution of the bootstrap sample mean of an infinitesimal array. Ann. Inst. H. Poincaré Probab. Statist. 34 23-48.
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.
15. DEL BARRIO, E. and MATRÁN, C. (2000). The weighted bootstrap mean for heavy-tailed distributions. J. Theoret. Probab. 13 547-569.
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.
17. DUDLEY, R. M. (1989). Real Analysis 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.
19. GINÉ, E. and ZINN, J. (1989). Necessary conditions for the bootstrap of the mean. Ann. Statist. 17 684-691.
20. GNEDENKO, B. V. and KOLMOGOROV, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA.
21. HÁJEK, J. (1961). Some extensions of the Wald-Wolfowitz-Noether theorem. Ann. Math. Statist. 32 506-523.
22. HÁJEK, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39 325-346.
23. HÁJEK, J., ŠIDÁK, Z. and SEN, P. K. (1999). Theory of Rank Tests, 2nd ed. Academic Press, Orlando, FL.
24. HALL, P. (1990). Asymptotic properties of the bootstrap for heavy-tailed distributions. Ann. Probab. 18 1342-1360.
25. HALL, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
26. HARTMAN, P. and WINTNER, A. (1942). On the infinitesimal generators of integral convolutions. Amer. J. Math. 64 273-298.
27. JANSSEN, A. (1989). Local asymptotic normality for randomly censored models with applications to rank tests. Statist. Neerlandica 43 109-125.
28. JANSSEN, A. (1991). Conditional rank tests for randomly censored data. Ann. Statist. 19 1434-1456.
29. JANSSEN, A. (1994). Sums of independent triangular arrays and extreme order statistics. Ann. Probab. 22 1766-1793.
30. JANSSEN, A. (1997). Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem. Statist. Probab. Lett. 36 9-21.
31. JANSSEN, A. (1999). Nonparametric symmetry tests for statistical functionals. Math. Methods Statist. 8 320-343.
32. JANSSEN, A. (2000). Invariance principles for sums of extreme sequential order statistics attracted to Lévy processes. Stochastic Process. Appl. 85 255-277.
33. JANSSEN, A. and MASON, D. M. (1990). Non-standard Rank Tests. Lecture Notes in Statist. 65. Springer, New York.
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.
35. KNIGHT, K. (1989). On the bootstrap of the sample mean in the infinite variance case. Ann. Statist. 17 1168-1175.
36. LEPAGE, R., WOODROOFE, M. and ZINN, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632.
37. MAMMEN, E. (1992a). When Does Bootstrap Work? Asymptotic Results and Simulations. Lecture Notes in Statist. 77. Springer, New York.
38. MAMMEN, E. (1992b). Bootstrap, wild bootstrap, and asymptotic normality. Probab. Theory Related Fields 93 439-455.
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.
40. MASON, D. M. and SHAO, Q.-M. (2001). Bootstrapping the Student t-statistic. Ann. Probab. 29 1435-1450.
41. NEUHAUS, G. (1988). Asymptotically optimal rank tests for the two-sample problem with randomly censored data. Comm. Statist. Theory Methods 17 2037-2058.
42. NEUHAUS, G. (1993). Conditional rank tests for the two-sample problem under random censorship. Ann. Statist. 21 1760-1779.
43. PETROV, V. V. (1995). Limit Theorems of Probability Theory. Clarendon Press, Oxford.
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.
45. ROMANO, J. P. (1989). Bootstrap and randomization tests of some nonparametric hypotheses. Ann. Statist. 17 141-159.
46. SHORACK, G. R. (1996). Linear rank statistics, finite sampling, permutation tests and winsorizing. Ann. Statist. 24 1371-1385.
47. SHORACK, G. R. (2000). Probability for Statisticians. Springer, New York.
48. STRASSER, H. and WEBER, C. (1999). The asymptotic theory of permutation statistics. Math. Methods Statist. 8 220-250.
49. TUCKER, H. G. (1975). The supports of infinitely divisible distribution functions. Proc. Amer. Math. Soc. 49 436-440.
50. VAN DER VAART, A. W. and WELLNER, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.
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.
52. WITTING, H. and NÖLLE, G. (1970). Angewandte Mathematische Statistik. Optimale finite und asymptotische Verfahren. Teubner, Stuttgart.
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.