Dimension comparison for Student's statistic under symmetric unimodality

Abstract

Let $q_n(r)$ denote the tail probability at $r$ of the self-normalized sum of $n$ independent centered uniform variables. At $r=3$, the first distribution-sensitive term in the two-sided Edgeworth expansion of Student's statistic vanishes. We evaluate the expansion at the common moving boundary $r_n=3+\lambda /n$ in dimensions $n$ and $n-k$. Uniformly over deletion ranks retaining a fixed positive fraction of observations, the first nonzero difference converges to an explicit phase surface $H_{\delta }(\lambda )$; its zero curve unifies fixed, sublinear and fixed-fraction deletions, with tangent crossing $12/35$. Through the Khintchine scale-mixture representation, this comparison yields a single compactly supported $C^{\infty }$ symmetric unimodal parent, independent of $n$, whose Student tail exceeds the equal-scale uniform tail for all sufficiently large $n$ along nominal levels tending to $2\{1-\Phi (\sqrt{3})\}$ from below. In contrast, a quantile-ratio order shows that the uniform parent maximizes every even moment and every convergent even power series with nonnegative coefficients. We also derive the fixed-confidence dimension expansion and an exact reversal at $n=6$.

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