Estimation of smooth functionals of covariance operators: jackknife bias reduction and bounds in terms of effective rank

by   Vladimir Koltchinskii, et al.

Let E be a separable Banach space and let X, X_1,…, X_n, … be i.i.d. Gaussian random variables taking values in E with mean zero and unknown covariance operator Σ: E^∗↦ E. The complexity of estimation of Σ based on observations X_1,…, X_n is naturally characterized by the so called effective rank of Σ: r(Σ):= 𝔼_ΣX^2/Σ, where Σ is the operator norm of Σ. Given a smooth real valued functional f defined on the space L(E^∗,E) of symmetric linear operators from E^∗ into E (equipped with the operator norm), our goal is to study the problem of estimation of f(Σ) based on X_1,…, X_n. The estimators of f(Σ) based on jackknife type bias reduction are considered and the dependence of their Orlicz norm error rates on effective rank r(Σ), the sample size n and the degree of Hölder smoothness s of functional f are studied. In particular, it is shown that, if r(Σ)≲ n^α for some α∈ (0,1) and s≥1/1-α, then the classical √(n)-rate is attainable and, if s> 1/1-α, then asymptotic normality and asymptotic efficiency of the resulting estimators hold. Previously, the results of this type (for different estimators) were obtained only in the case of finite dimensional Euclidean space E=ℝ^d and for covariance operators Σ whose spectrum is bounded away from zero (in which case, r(Σ)≍ d).


page 1

page 2

page 3

page 4


Estimation of Smooth Functionals in Normal Models: Bias Reduction and Asymptotic Efficiency

Let X_1,..., X_n be i.i.d. random variables sampled from a normal distri...

Efficient Estimation of Smooth Functionals in Gaussian Shift Models

We study a problem of estimation of smooth functionals of parameter θ o...

Asymptotically Efficient Estimation of Smooth Functionals of Covariance Operators

Let X be a centered Gaussian random variable in a separable Hilbert spac...

The spatial sign covariance operator: Asymptotic results and applications

Due to the increasing recording capability, functional data analysis has...

Functional estimation in log-concave location families

Let {P_θ:θ∈ℝ^d} be a log-concave location family with P_θ(dx)=e^-V(x-θ)d...

Continuously Indexed Graphical Models

Let X = {X_u}_u ∈ U be a real-valued Gaussian process indexed by a set U...

Robust Modifications of U-statistics and Applications to Covariance Estimation Problems

Let Y be a d-dimensional random vector with unknown mean μ and covarianc...

Please sign up or login with your details

Forgot password? Click here to reset