Sharp Risk Bounds and a Shrinking-Variance Transition for Normal and Gamma Fits
Year: 2026
Type: preprint
Abstract: Suppose lognormal data are fitted by an ordinary normal model and a shape-scale gamma model. If s > 0 is the log variance, the difference Delta(s) between their optimized forward Kullback-Leibler risks satisfies 2s/3 < Delta(s) < s and 2/3 < Delta'(s) < 1; both constants are optimal. We establish the boundary behavior of the maximized gamma-to-normal log-likelihood ratio R_n when s = s_n tends to zero. If n s_n tends to c in (0, infinity), R_n converges in distribution to N(2c/3, 2c/3). If n s_n tends to infinity, gamma is selected with probability tending to one; if n s_n tends to zero, that probability tends to one half. An exact profile identity and uniform sample-moment expansion give the result. Fixed-variance likelihood and compact-prior marginal asymptotics are also included. The KL projection formulas and general misspecification theory are established prior ingredients. This is a scoped response to AIM Singular learning theory Problem 1.3, indexed as AIM-ALGEBRAIC_GEOMETRY-0271 in UnsolvedMath v1.6.0, not a resolution of every interpretation of the source prompt. The preprint is unrefereed and makes no absolute priority claim.
Author: Ferudun, Alper
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222250656
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