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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.

MSC:

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

Citations:

Zbl 0958.60023
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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 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
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