Cramér-Rao lower bound and information geometry

Author

Nielsen, Frank

Abstract

This article focuses on an important piece of work of the world renowned Indian statistician, Calyampudi Radhakrishna Rao. In 1945, C. R. Rao (25 years old then) published a pathbreaking paper [43], which had a profound impact on subsequent statistical research. Roughly speaking, Rao obtained a lower bound to the variance of an estimator. The importance of this work can be gauged, for instance, by the fact that it has been reprinted in the volume Breakthroughs in Statistics: Foundations and Basic Theory [32]. There have been two major impacts of this work: First, it answers a fundamental question statisticians have always been interested in, namely, how good can a statistical estimator be? Is there a fundamental limit when estimating statistical parameters? Second, it opens up a novel paradigm by introducing differential geometric modeling ideas to the field of Statistics. In recent years, this contribution has led to the birth of a flourishing field of Information Geometry [6].