Bias Reduction and Asymptotic Efficiency in Estimation of Smooth Functionals of High-Dimensional Covariance

Abstract: We discuss a recent approach to bias reduction in a problem of estimation of smooth functionals of high-dimensional parameters of statistical models. In particular, this approach has been developed in the case of estimation of functionals of covariance operator Σ : Rd ^d → R^d of the form f(Σ), B based on n i.i.d. observations X₁, . . . , X_n sampled from the normal distribution with mean zero and covariance Σ, f : R → R being a sufficiently smooth function and B being an operator with nuclear norm bounded by a constant. This includes such problems as estimation of bilinear forms (for instance, matrix entries in a given basis) of spectral projections of unknown covariance that are of importance in principal component analysis. A “bootstrap chain” bias reduction method, based on an approximate solution of a certain integral equation (the Wishart equation) on the cone of self-adjoint positive semidefinite operators, yields asymptotically efficient estimators of the functional f(Σ), B under proper assumptions on the growth of dimension d and smoothness of function f. In particular, this holds under the assumption that d ≤ n^α for some α ∈ (0, 1) and that f belongs to a Besov space B^s_∞,1(R) for s > ¹ . The proof of asymptotic efficiency relies on a number of probabilistic and analytic tools (operator differentiability; Gaussian concentration; properties of Wishart operators and orthogonally invariant functions on the cone of positive semidefinite operators; information-theoretic lower bounds).

Biography: Vladimir Koltchinskii is a professor in Mathematics at Georgia Tech. His current research is primarily in high-dimensional statistics and probability.