Operator Geometry in Statistics
This article discusses the essentials of operator trigonometry developed by the author as it applies to statistics, with emphasis on key elements such as operator antieigenvalues, operator antieigenvectors, and operator turning angles. Operator trigonometry started out infinite dimensional, and remains infinite dimensional, even for Banach spaces. Thus, it is in principle applicable not only to infinite-dimensional statistics but also to cases involving functional data. The article first considers how operator trigonometry gives new geometrical meaning to statistical efficiency before formalizing it in a more deductive manner. It then explains the essentials of operator trigonometry and summarizes the ensuing developments. It also describes two lemmas that are implicit and essential to operator trigonometry, Antieigenvector Reconstruction Lemma and General Two-Component Lemma, and how operator trigonometry provides new geometry to statistics matrix inequalities and canonical correlations. Finally, it presents new results applying operator trigonometry to prediction theory and to association measures.