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

Mason, David M.; Shao, Qi-Man

Ann. Probab. 29, No. 4, 1435-1450 (2001).

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

Keywords:

bootstrap; Student t-statistic; order statistics

Citations:

Zbl 0958.60023

References:

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