Abstract

This paper presents a comprehensive geometric study of the statistical manifold of the two-parameter Shanker and Mishra Lindley Distribution (S.M.L.D). It first proves that the distribution family forms a smooth two-dimensional statistical manifold parameterised by  with injective parameter mappings. It then establishes the foundational structure by deriving the Riemannian Fisher metric. , the geodesic equations using Christoffel symbols, and the Hellinger distance between any two distributions. The paper further explores the concept of -connections and proves the existence of a unique flattening coefficient  that transforms the manifold from being curved under the exponential connection into a flat manifold. Finally, it examines the special case of fixing the parameter. , which reduces the model to a one-parameter exponential family. For this submanifold, the dual coordinates , the corresponding Fisher metric, the geodesic equations in dual coordinates, and the associated canonical (Bregman) divergence are derived. It aims to provide researchers with advanced tools (such as geometric statistical measures or information geometry) to enhance the accuracy and efficiency of parametric statistical inference.