Abstract
In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the -connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the -connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.
Highlights
From the geometrical viewpoint, a parametric statistical model can be considered a differentiable manifold, and the parameter space can be regarded as a coordinate system of the manifold [1,2]
When the density function f (x; θ) is sufficiently smooth in θ and it is differentiable as a function of θ, it is natural to introduce the structure of an k-dimensional manifold in the statistical model F, where θ plays the role of a coordinate system
These results show that the predictive density function, when the sample size n approaches infinity, is the estimative density function in the asymptotic sense
Summary
A parametric statistical model can be considered a differentiable manifold, and the parameter space can be regarded as a coordinate system of the manifold [1,2]. When the density function f (x; θ) is sufficiently smooth in θ and it is differentiable as a function of θ, it is natural to introduce the structure of an k-dimensional manifold in the statistical model F , where θ plays the role of a coordinate system The geometrical quantities, such as connection, divergence, flatness, curvature and tangent space, play a fundamental role in the statistical inference and asymptotic theory (see, for example, Komaki [3,4] and Harsha and Moosath [5]). We investigate the tangent space, affine connection, α-connection, torsion and Riemann-Christoffel curvature of the manifold of the exponential family of distributions with progressive Type-II censoring scheme. These geometric quantities can be applied to different areas in statistics such as Bayesian analysis. Monte Carlo simulation results and a real data analysis are presented in Section 6 to illustrate the main results
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
