Higher-order nonlinear shrinkage estimator of large-dimensional precision matrix

This paper introduces a new type of nonlinear shrinkage estimators for the precision matrix in high-dimensional settings, where the dimension of the data-generating process exceeds the sample size. The proposed estimators incorporate the Moore–Penrose inverse and the ridge-type inverse of the sample covariance matrix, and they include linear shrinkage estimators as special cases. Recursive formulae of these higher-order nonlinear shrinkage estimators are derived using partial exponential Bell polynomials. Through simulation studies, the new methods are compared with the oracle nonlinear shrinkage estimator of the precision matrix for which no analytical expression is available.