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

Abstract

In this communication, we first compare zα and tν,α, the upper 100α% points of a standard normal and a Student’s tν distributions respectively. We begin with a proof of a well-known result, namely, for every fixed \(0 zα. Next, Theorem 3.1 provides a new and explicit expression bν( > 1) such that for every fixed \(0 bνzα. 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.