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A Cramér Type Large Deviation Result for Student's t-Statistic

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  • Qi-Man Shao

    (University of Oregon)

Abstract

Let X, X 1, X 2,... be independent and identically distributed random variables with a finite third moment, and let T n be the Student's t-statistic. This paper shows that lim n→∞ P(T n>x)/P(t n>x)=1 holds uniformly in 0≤x≤o(n 1/6), where t n has a t-distribution with n−1 degrees of freedom. An example is also given to show that a finite third moment is necessary for this result.

Suggested Citation

  • Qi-Man Shao, 1999. "A Cramér Type Large Deviation Result for Student's t-Statistic," Journal of Theoretical Probability, Springer, vol. 12(2), pages 385-398, April.
    • Handle: RePEc:spr:jotpro:v:12:y:1999:i:2:d:10.1023_a:1021626127372
    DOI: 10.1023/A:1021626127372
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    References listed on IDEAS

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    1. Xuming He & Qi-man Shao, 1996. "Bahadur efficiency and robustness of studentized score tests," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(2), pages 295-314, June.
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    Cited by:

    1. Qiying Wang, 2011. "Refined Self-normalized Large Deviations for Independent Random Variables," Journal of Theoretical Probability, Springer, vol. 24(2), pages 307-329, June.
    2. Xinwei Feng & Qi-Man Shao & Ofer Zeitouni, 2021. "Self-normalized Moderate Deviations for Random Walk in Random Scenery," Journal of Theoretical Probability, Springer, vol. 34(1), pages 103-124, March.
    3. Bing-Yi Jing & Qiying Wang & Wang Zhou, 2015. "Cramér-Type Moderate Deviation for Studentized Compound Poisson Sum," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1556-1570, December.
    4. Han-Ying Liang & Josef G. Steinebach, 2006. "Self-Normalized LIL for Hanson–Russo Type Increments," Journal of Theoretical Probability, Springer, vol. 19(1), pages 70-88, January.

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