On a Sharper Lower Bound for a Percentile of a Student’s t Distribution with an Application

In this communication, we first compare z α and t ν,α, the upper 100α% points of a standard normal and a Student’s t ν distributions respectively. We begin with a proof of a well-known result, namely, for every fixed $0 z α . Next, Theorem 3.1 provides a new and explicit expression b ν ( > 1) such that for every fixed $0 b ν z α . This is clearly a significant improvement over the result that is customarily quoted in nearly every textbook and elsewhere. A proof of Theorem 3.1 is surprisingly simple and pretty. We also extend Theorem 3.1 in the case of a non-central Student’s t distribution (Section 3.3). In the context of Stein’s (Ann Math Stat 16:243–258, 1945; Econometrica 17:77–78, 1949) 100(1 − α)% fixed-width confidence intervals for the mean of a normal distribution having an unknown variance, we have examined the oversampling rate on an average for a variety of choices of m, the pilot sample size. We ran simulations to investigate this issue. We have found that the oversampling rates are approximated well by $t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2}$ for small and moderate values of m( ≤ 50) all across Table 2 where ν = m − 1. However, when m is chosen large (≥ 100), we find from Table 3 that the oversampling rates are not approximated by $t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2}$ very well anymore in some cases, and in those cases the oversampling rates either exceed the new lower bound of $t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2},$ namely $b_{\nu }^{2},$ or comes incredibly close to $b_{\nu }^{2}$ where ν = m − 1. That is, the new lower bound for a percentile of a Student’s t distribution may play an important role in order to come up with diagnostics in our understanding of simulated output under Stein’s fixed-width confidence interval method.