Fisher-type information involving higher order derivatives

TL;DR

The paper systematically develops Fisher-type information based on higher-order derivatives $I^{(p)}$ and relative variants, establishing foundational regularity, integrability, decay, and isoperimetric representations. It proves key properties: lower semi-continuity under weak convergence, convexity under mixtures, and monotonicity/continuity under Gaussian smoothing; provides Stam-type inequalities for $p\ge2$, including Gaussian-component refinements. Using isoperimetric profiles, it derives links between $I$, $I^{(2)}$, and density shape, and obtains lower bounds like $I^{(2)}(X) \ge \tfrac{1}{3} I_4(X) \ge \tfrac{1}{3} I(X)^2$, with conditions guaranteeing finiteness. The Gamma and other explicit distributions illustrate finiteness thresholds, sharp inequalities, and the limitations of stronger cross-term bounds, highlighting both the reach and the boundaries of higher-order Fisher-type information in convergence and entropy-like analyses.

Abstract

Basic general properties are considered for the Fisher-type information involving higher order derivatives. They are used to explore various properties of probability densities and to derive Stam-type inequalities.