A finite coefficient theory for shape identification in tilted generalized gamma families

Abstract

A continuous shape parameter enters both a radial density and the normalization of its transform coefficients. This coupling allows changes in shape to be compensated by changes in scale and exponential tilt. We study the coefficients b_k=e^a E[Y^k ]/[k!(d)_k ] for densities proportional to y^(d-1) exp(-αy^q+\b{eta}y), with known integer q>=2, d,α>0, and unrestricted real a,\b{eta}. For q=2, the first four coefficients determine all parameters globally, whereas three coefficients leave an analytic curve of solutions. We characterize the exact image of the two shape ratios and prove that two alternating corners decide whether an entire coefficient box is realizable and give its exact shape range. For every fixed q, the first q(q+1) coefficients give global identification and an everywhere full-rank map. A finite-difference criterion yields generic identification from 2q+1 coefficients, including every zero-tilt model against the full tilted family. For q=3, an exact rational certificate constructs two distinct shapes with identical first four coefficients and separated fifth coefficients. The results distinguish exact identification, realizability, and sensitivity for a parameter-dependent coefficient normalization.