Bootstrapping the Student \(t\)-statistic. (English) Zbl 1010.62026

Summary: Let \(X_1,\dots, X_n\), \(n\geq 1\), be independent, identically distributed random variables and consider the Student \(t\)-statistic \(T_n\) based upon these random variables. E. Giné, F. Götze and D.M. Mason [ibid. 25, No. 3, 1514-1531 (1997; Zbl 0958.60023)] proved that \(T_n\) converges in distribution to a standard normal random variable if and only if \(X\) is in the domain of attraction of a normal random variable and \(EX=0\). We shall show that roughly the same holds true for the bootstrapped Student \(t\)-statistic \(T^*_n\). In the process we shall disclose all the possible subsequential limiting laws of \(T^*_n\). The proofs introduce a number of amusing tricks that may be of independent interest.


62F40 Bootstrap, jackknife and other resampling methods
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems


Zbl 0958.60023
Full Text: DOI


[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 correction to ”The bootstrap of the mean with arbitrary bootstrap sample size.” Ann. Inst. H. Poincaré Probab. Statist. 27 583- 595. · Zbl 0747.62019
[4] Athreya, K. B. (1987). Bootstrap of the mean in the infinite variance case. Ann. Statist. 15 724-731. · Zbl 0628.62042
[5] Chen, J. and Rubin, H. (1984). A note on the behavior of sample statistics when the population mean is infinite. Ann. Probab. 12 256-261. · Zbl 0551.60034
[6] Cs örg o, S. and Mason, D. M. (1989). Bootstrapping empirical functions. Ann. Statist. 17 1447-1471. · Zbl 0701.62057
[7] 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
[8] Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics, 2nd. ed. Krieger, Malabar, FL. · Zbl 0634.62044
[9] Giné, E., G ötze, F. and Mason, D. M. (1997). When is the Student t-statistic asymptotically standard normal? Ann. Probab. 25 1514-1531. · Zbl 0958.60023
[10] Giné, E. and Zinn, J. (1989). Necessary conditions for the bootstrap of the mean. Ann. Statist. 17 684-691. · Zbl 0672.62026
[11] Hall, P. (1990). Asymptotic properties of the bootstrap for heavy-tailed distributions. Ann. Probab. 18 1342-1360. · Zbl 0714.62035
[12] Maller, R. A. and Resnick, S. I. (1984). Limiting behaviour of sums and the term of maximum modulus. Proc. London Math. Soc. 49 385-422. · Zbl 0525.60036
[13] O’Brien, G. L. (1980). A limit theorem for sample maxima and heavy branches in GaltonWatson trees. J. Appl. Probab. 17 539-545. JSTOR: · Zbl 0428.60034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.