## 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

### Keywords:

bootstrap; Student t-statistic; order statistics

Zbl 0958.60023
Full Text:

### References:

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