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**Saddlepoint approximation for Student’s \(t\)-statistic with no moment conditions.**
*(English)*
Zbl 1068.62016

Summary: A saddlepoint approximation of the Student’s \(t\)-statistic was derived by H. E. Daniels and G. A. Young [Biometrika 78, 169–179 (1991)] under the very stringent exponential moment condition that requires that the underlying density function goes down at least as fast as a normal density in the tails. This is a severe restriction on the approximation applicability.

We show that this strong exponential moment restriction can be completely dispensed with, that is, saddlepoint approximation of the Student’s \(t\)-statistic remains valid without any moment condition. This confirms the folklore that the Student’s \(t\)-statistic is robust against outliers. The saddlepoint approximation not only provides a very accurate approximation for the Student’s \(t\) statistic, but it also can be applied much more widely in statistical inference. As a result, saddlepoint approximations should always be used whenever possible. Some numerical work will be given to illustrate these points.

We show that this strong exponential moment restriction can be completely dispensed with, that is, saddlepoint approximation of the Student’s \(t\)-statistic remains valid without any moment condition. This confirms the folklore that the Student’s \(t\)-statistic is robust against outliers. The saddlepoint approximation not only provides a very accurate approximation for the Student’s \(t\) statistic, but it also can be applied much more widely in statistical inference. As a result, saddlepoint approximations should always be used whenever possible. Some numerical work will be given to illustrate these points.

### MSC:

62E20 | Asymptotic distribution theory in statistics |

### Keywords:

large deviation; asymptotic normality; Edgeworth expansions; self-normalized sum; Student’s \(t\)-statistic; absolute error; relative error### Software:

bootlib### References:

[1] | Daniels, H. E. and Young, G. A. (1991). Saddlepoint approximation for the Studentized mean, with an application to the bootstrap. Biometrika 78 169–179. |

[2] | Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Univ. Press. · Zbl 0886.62001 |

[3] | Dembo, A. and Shao, Q.-M. (1998). Self-normalized large deviations in vector spaces. In High-Dimensional Probability (E. Eberlein, M. Hahn and M. Talagrand, eds.) 27–32. Birkhäuser, Basel. · Zbl 0910.60011 |

[4] | Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2 , 2nd ed. Wiley, New York. · Zbl 0219.60003 |

[5] | Field, C. and Ronchetti, E. (1990). Small Sample Asymptotics. IMS, Hayward, CA. · Zbl 0742.62016 |

[6] | Giné, E., Götze, F. and Mason, D. M. (1997). When is the Student \(t\)-statistic asymptotically standard normal? Ann. Probab. 25 1514–1531. · Zbl 0958.60023 · doi:10.1214/aop/1024404523 |

[7] | Griffin, P. S. and Kuelbs, J. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab. 17 1571–1601. JSTOR: · Zbl 0687.60033 · doi:10.1214/aop/1176991175 |

[8] | Griffin, P. S. and Kuelbs, J. (1991). Some extensions of the LIL via self-normalizations. Ann. Probab. 19 380–395. JSTOR: · Zbl 0722.60028 · doi:10.1214/aop/1176990551 |

[9] | Hall, P. (1987). Edgeworth expansion for Student’s \(t\) statistic under minimal moment conditions. Ann. Probab. 15 920–931. JSTOR: · Zbl 0626.62019 · doi:10.1214/aop/1176992073 |

[10] | Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Univ. Press. · Zbl 1274.62008 |

[11] | Kolassa, J. E. (1997). Series Approximation Methods in Statistics , 2nd ed. Lecture Notes in Statist. 88 . Springer, New York. · Zbl 0877.62013 |

[12] | Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab. 1 788–809. · Zbl 0272.60016 · doi:10.1214/aop/1176996846 |

[13] | Lugannani, R. and Rice, S. (1980). Saddlepoint approximation for the distribution of the sum of independent random variables. Adv. in Appl. Probab. 12 475–490. · Zbl 0425.60042 · doi:10.2307/1426607 |

[14] | Reid, N. (1988). Saddlepoint methods and statistical inference (with discussion). Statist. Sci. 3 213–238. · Zbl 0955.62541 |

[15] | Shao, Q.-M. (1997) Self-normalized large deviations. Ann. Probab. 25 285–328. · Zbl 0873.60017 · doi:10.1214/aop/1024404289 |

[16] | Wang, Q. Y. and Jing, B.-Y. (1999). An exponential nonuniform Berry–Esseen bound for self-normalized sums. Ann. Probab. 27 2068–2088. · Zbl 0972.60011 · doi:10.1214/aop/1022677562 |

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