Student’s t-Statistic under Unimodal Densities

Abstract

Consider an unknown distribution with a symmetric unimodal density and the induced location-scale family. We study confidence intervals for the location parameter based on Student’s t-statistic, and we conjecture that the uniform distribution is least favorable in that it leads to confidence intervals that are largest given their coverage probability, provided the nominal confidence level is large enough. This conjecture is supported by an argument based on second order asymptotics in the sample size and on asymptotics in the length of the confidence interval, by a finite sample inequality, and by simulation results.