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

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