A new asymptotic representation and inversion method for the Student's t distribution

Abstract

Some special functions are particularly relevant in applied probability and statistics. For example, the incomplete beta function is the cumulative central beta distribution. In this paper, we consider the inversion of the central Student's-t distribution which is a particular case of the central beta distribution. The inversion of this distribution function is useful in hypothesis testing as well as for generating random samples distributed according to the corresponding probability density function. A new asymptotic representation in terms of the complementary error function will be one of the important ingredients in our analysis. As we will show, this asymptotic representation is also useful in the computation of the distribution function. We illustrate the performance of all the obtained approximations with numerical examples.

Keywords:

• Incomplete beta function
• asymptotic expansions
• Student's-t distribution

2020 Mathematics Subject Classifications:

• 33B20
• 41A60
• 65D20

Acknowledgments

We thank the reviewer for his/her constructive and helpful remarks. NMT thanks CWI, Amsterdam, for scientific support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Financial support from Ministerio de Ciencia e Innovación, Spain, project PGC2018-098279-B-I00 (MCIN/AEI/10.13039/501100011033/FEDER) is acknowledged.