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


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