Information geometry on the curved q-exponential family with application to survival data analysis

Information geometry and statistical manifold have attracted wide attention in the past few decades. The Amari-Chentsov structure plays a central role both in optimization and statistical inference because of the invariance under sufficient statistics. This paper discusses the Amari-Chentsov structure on statistical manifolds induced by the curved q-exponential family with Type-I censored data. The reliability function and cumulant generating function of the family are obtained, where the cumulant generating function contains a random variable depends on the experimental design. After the construction of the statistical manifold, the geometric quantities, i.e., Fisher metric, Amari tensor and connection coefficients are investigated. We find that the affine connection is not equal to 0, which is different from the existing results. The relationship among the geometric quantities based on the q-exponential family and curved q-exponential family is discussed. The relationship between the curvature on the manifold in the normal and tangent directions is obtained. Employing the maximum likelihood and Bayesian methods, the parameter vector and its dual are estimated. The main results are illustrated with the two-parameter q-exponential distribution and a real data analysis based on a terminal human cancer data set.