The asymptotic variance of the univariate PLS estimator

Abstract

In this short note, we derive an expression for the asymptotic covariance matrix of the univariate partial least squares (PLS) estimator. In contrast to M.C. Denham [J. Chemometrics 11 (1997) 39], who provided a locally linear approximation based on a recursive definition of the estimator, we derive a more compact expression for the asymptotic covariance matrix by combining a standard convergence result with matrix differential calculus, in particular the approach of J.R. Magnus and H. Neudecker [Matrix Differential Calculus with Applications in Statistics and Econometrics, revised ed., Wiley, Chichester, UK, 1991]. We also describe some theoretical and practical aspects of calculating the asymptotic covariance matrix, and illustrate its use on spectroscopic data.