Shun-ichi Amari 
Differential-Geometrical Methods in Statistics [PDF ebook] 

From the reviews: "In this Lecture Note volume the author describes his differential-geometric approach to parametrical statistical problems summarizing the results he had published in a series of papers in the last five years. The author provides a geometric framework for a special class of test and estimation procedures for curved exponential families. … … The material and ideas presented in this volume are important and it is recommended to everybody interested in the connection between statistics and geometry …" #Metrika#1 "More than hundred references are given showing the growing interest in differential geometry with respect to statistics. The book can only strongly be recommended to a geodesist since it offers many new insights into statistics on a familiar ground." #Manuscripta Geodaetica#2