High Relative Accuracy Computations With Covariance Matrices of Order Statistics

ABSTRACT In many statistical applications, numerical computations with covariance matrices need to be performed. The error made when performing such numerical computations increases with the condition number of the covariance matrix, which is related to the number of variables and the strength of the correlation between the variables. In a recent work, a method for estimating the covariance matrix of a Gaussian Markov Random Field under a total positivity constraint was proposed. This estimation allows for performing many numerical computations with covariance matrices to high relative accuracy (the relative error is of the order of machine precision). However, the necessary conditions for this estimation method to produce a covariance matrix that is close to the population covariance matrix may be too demanding for real‐life data. In this paper, we study a particular setting related to order statistics in which these necessary conditions are inherently satisfied. In addition to the theoretical study, an extensive discussion concerning many potential applications is addressed, and a real‐life example of an application related to sports data is presented.