Abstract
We introduce the notion of a C k -diffeological statistical model, which allows us to apply the theory of diffeological spaces to (possibly singular) statistical models. In particular, we introduce a class of almost 2-integrable C k -diffeological statistical models that encompasses all known statistical models for which the Fisher metric is defined. This class contains a statistical model which does not appear in the Ay–Jost–Lê–Schwachhöfer theory of parametrized measure models. Then, we show that, for any positive integer k , the class of almost 2-integrable C k -diffeological statistical models is preserved under probabilistic mappings. Furthermore, the monotonicity theorem for the Fisher metric also holds for this class. As a consequence, the Fisher metric on an almost 2-integrable C k -diffeological statistical model P ⊂ P ( X ) is preserved under any probabilistic mapping T : X ⇝ Y that is sufficient w.r.t. P. Finally, we extend the Cramér–Rao inequality to the class of 2-integrable C k -diffeological statistical models.
Highlights
In mathematical statistics, the notion of a statistical model and the notion of a parameterized statistical model are of central importance [1]
If the parameter set Θ is a smooth manifold, we can study a statistical model p(Θ), endowed with a parameterization p : Θ → p(Θ) ⊂ P (X ), by applying differential geometric techniques to Θ and to smooth the mappings p : Θ → P (X ). This idea lies in the heart of the field of information geometry, which is in the domain of mathematical statistics, where we study statistical models using techniques of differential geometry [2,3,4,5]
We extend conceptually many results in the Ay–Jost–Lê–Schwachhöfer theory concerning the differential geometry of parameterized statistical models and their application to statistics and to the class of
Summary
The notion of a statistical model and the notion of a parameterized statistical model are of central importance [1]. According to currently accepted theories, see e.g., [1] and the references therein, a statistical model is a subset PX ⊂ P (X ) and a parameterized statistical model is a parameter set Θ, together with a mapping p : Θ → P (X ). If the parameter set Θ is a smooth manifold, we can study a statistical model p(Θ), endowed with a parameterization p : Θ → p(Θ) ⊂ P (X ), by applying differential geometric techniques to Θ and to smooth the mappings p : Θ → P (X ). The theory of parameterized measure models, allows us to study singular statistical models PX using differential geometric techniques, if PX is endowed with a parameterization by a Banach manifold. The class of C1 -diffeological statistical models is larger than the class of statistical models parameterized by Banach manifolds as the Ay–Jost–Lê–Schwachhöfer theory. We conclude our paper with a discussion on some future directions and open questions
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