Anti-Modes, Distributional Gaps, and the Fisher Information Profile: Nonparametric Estimation and Inference

<div> We develop a nonparametric theory of distributional gaps centred on the Fisher information profile where Q is the quantile function and f the density. A percentile p * is an anti-mode percentile if and only if it is simultaneously a local minimiser of f , a local maximiser of the quantile density q(p) = Q ′ (p), and a local maximiser of the asymptotic quantile estimation variance σ 2 Q (p)-and, under a mild first-order condition, a local minimiser of I(p)-providing the first information-theoretic characterisation of distributional gaps. We prove n 2/5 consistency and asymptotic normality of the kernel plug-in estimator p * , uniform convergence of Î(p) at rate Op((nh) -1/2 + h 2 ), and bootstrap size control with nontrivial power against local alternatives shrinking at rate n -2/5 . Monte Carlo experiments across twelve data-generating processes confirm the theoretical rates, and an application to Old Faithful geyser data illustrates the full inferential output. </div>