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

Qi Man Shao

doi:10.1023/A:1021626127372

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

Qi Man Shao^*

^*Corresponding author for this work

Research output: Contribution to journal › Journal Article › peer-review

68 Citations (Scopus)

Abstract

Let X, X₁, X₂,... 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.

Original language	English
Pages (from-to)	385-398
Number of pages	14
Journal	Journal of Theoretical Probability
Volume	12
Issue number	2
DOIs	https://doi.org/10.1023/A:1021626127372
Publication status	Published - 1999
Externally published	Yes

Keywords

Asymptotic distribution
Self-normalized large deviation
t-statistic

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10.1023/A:1021626127372

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title = "A Cram{\'e}r Type Large Deviation Result for Student's t-Statistic",

abstract = "Let X, X1, X2,... be independent and identically distributed random variables with a finite third moment, and let Tn be the Student's t-statistic. This paper shows that limn → ∞ P(Tn > x)/P(tn > x) = 1 holds uniformly in 0 ≤ x ≤ o(n1/6), where tn 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.",

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N2 - Let X, X1, X2,... be independent and identically distributed random variables with a finite third moment, and let Tn be the Student's t-statistic. This paper shows that limn → ∞ P(Tn > x)/P(tn > x) = 1 holds uniformly in 0 ≤ x ≤ o(n1/6), where tn 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.

AB - Let X, X1, X2,... be independent and identically distributed random variables with a finite third moment, and let Tn be the Student's t-statistic. This paper shows that limn → ∞ P(Tn > x)/P(tn > x) = 1 holds uniformly in 0 ≤ x ≤ o(n1/6), where tn 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.

KW - Asymptotic distribution

KW - Self-normalized large deviation

KW - t-statistic

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UR - https://www.scopus.com/pages/publications/0033441791

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DO - 10.1023/A:1021626127372

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SP - 385

EP - 398

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 2

ER -

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

Abstract

Keywords

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