An Invitation to Operator-Based Statistics

This article deals with operator-based statistics and its advantages. It first provides an overview of the historical and pedagogical aspects of operator-based statistics before explaining the underlying practical and theoretical motivations, along with synthetic and conceptual arguments. In particular, it develops the operator-based approach for factor multivariate analysis (and for their asymptotic studies) and offers several examples that show the value of operators in statistics. The discussion focuses on covariance operators, Hankel and Toeplitz operators, regression operators, measure-associated operators, tensor operators, and some other important categories. The article also describes noncommutative or quantum statistics and concludes with some reflections on the key notions and formulations of a "unified statistics" and projectors in (classical) statistics.