On statistics which are almost sufficient from the viewpoint of the Fisher metrics (Q6660918)

This paper introduces and studies a quantitative version of sufficient statistics such that the Fisher metric of the induced model is bi-Lipschitz equivalent to that of the original model. The principal result of the paper is a characterization of such statistics in terms of density functions that are parallel to [\textit{N. Ay} et al., Information geometry. Cham: Springer (2017; Zbl 1383.53002)].\N\N\textit{S.-i. Amari} and \textit{H. Nagaoka} [Methods of information geometry. Transl. from the Japanese by Daishi Harada. Providence, RI: AMS, American Mathematical Society; Oxford: Oxford University Press (2000; Zbl 0960.62005)] characterized sufficcient statistics in terms of the Fisher metrics, which measures the informational difference of measures in the statistical model. \textit{N. Ay} et al. [Bernoulli 24, No. 3, 1692--1725 (2018; Zbl 1419.62057); Information geometry. Cham: Springer (2017; Zbl 1383.53002)] introduced the framework of parametrized measure models to rigorously formalize the statistical models with infinite sample spaces, redefining sufficient statistics in the context and giving its characterization in terms of density functions to generalize the result of Amari-Nagaoka.\N\NSufficient statistics can be used to improve estimates on the statistical model by theorems of \textit{D. Blackwell} [Ann. Math. Stat. 18, 105--110 (1947; Zbl 0033.07603)], \textit{A. N. Kolmogorov} [Izv. Akad. Nauk SSSR, Ser. Mat. 14, 303--326 (1950; Zbl 0039.15102); Am. Math. Soc., Transl. 98, 28 p. (1950; Zbl 0052.36804)] and \textit{C. R. Rao} [Bull. Calcutta Math. Soc. 37, 81--91 (1945; Zbl 0063.06420)] and \textit{E. L. Lehmann} and \textit{H. Scheffé} [Sankhyā 10, 305--340 (1950; Zbl 0041.46301); Sankhyā 15, 219--236 (1955; Zbl 0068.12907)].