At random points of the integer lattice we place boxes of random sizes. The question we ask is under what conditions the entire space is covered. Also if instead of the entire lattice, our underlying space is a subset of it, then what can we say about the coverage properties.

Differential geometric approach to the field of statistics gave rise to a branch of mathematics called the information geometry. Information geometry began as a study of the geometric structures possessed by a statistical model of probability distributions. A statistical model equipped with a Riemannian metric together with a pair of dual affine connections is called a statistical manifold. Information geometry typically deals with the study of various geometric structures on a statistical manifold. In this talk I present a brief description of the information geometric framework for the statistical estimation problem. First I describe two important class of geometric structures on a statistical manifold, the alpha-geometry and the ( F, G)-geometry. Then the role of the (F, G)-geometry in the study of dually flat structures of the deformed exponential family is discussed. Also I describe the geometric framework for the mismatched estimation problem in an exponential family. Finally I present some of the open research problems in the area of estimation in a deformed exponential family