On Asymptotic and Strict Monotonicity of a Sharper Lower Bound for Student’s t Percentiles

Abstract

Let z _α and t _ν,α denote the upper 100α% points of a standard normal and a Student’s t _ν distributions respectively. It is well-known that for every fixed $0<\alpha <\frac{1}{2}$ and degree of freedom ν, one has t _ν,α > z _α and that t _ν,α monotonically decreases to z _α as ν increases. Recently, Mukhopadhyay (Methodol Comput Appl Probab, 2009) found a new and explicit expression b _ν ( > 1) such that t _ν,α > b _ν z _α for every fixed $0<\alpha <\frac{1}{2}$ and ν. He also showed that b _ν converges to 1 as ν increases. In this short note, we prove three key results: (i) $\log(b_{\nu+1}/b_{\nu})\sim -\frac{1}{4}\nu^{-2}$ for large enough ν, (ii) b _ν strictly decreases as ν increases, and (iii) $b_{\nu}\sim 1+\frac14\nu^{-1}+\frac1{32}\nu^{-2}$ for large enough ν.