Estimation of trace functionals and spectral measures of covariance operators in Gaussian models

Let f: R₊↦ R be a smooth function with $f(0)=0.$ A problem of estimation of a functional τf(Σ):= tr(f(Σ)) of unknown covariance operator Σ in a separable Hilbert space H based on i.i.d. mean zero Gaussian observations X₁,, Xₙ with values in H and covariance operator Σ is studied. Let Σₙ be the sample covariance operator based on observations X₁,, Xₙ. Estimators {align*} Tf,m(X_1,, X_n):= ∑ⱼ₌₁^m C_j τ_f( Σn_j) {align*} based on linear aggregation of several plug-in estimators τf( Σnⱼ), where the sample sizes n/c≤ n₁<<nₘ≤ n and coefficients C₁,, Cₙ are chosen to reduce the bias, are considered. The complexity of the problem is characterized by the effective rank r(Σ):= tr(Σ)\|Σ\| of covariance operator Σ. It is shown that, if f∈ Cᵐ⁺¹( R₊) for some m≥ 2, \|f''\|_L∞ 1, \|f⁽ᵐ⁺¹⁾\|_L∞ 1, \|Σ\| 1 and r(Σ) n, then {align*} & \| Tf,m(X_1,, X_n)-τ_f(Σ)\|L_2 _m {\|Σ f'(Σ)\|_2}{√n} + {{ r}(Σ)}{n}+ { r}(Σ)(√{{{ r}(Σ)}{n}})ᵐ⁺¹. {align*} Similar bounds have been proved for the Lₚ-errors and some other Orlicz norm errors of estimator Tf,m(X₁,, Xₙ). The optimality of these error rates, other estimators for which asymptotic efficiency is achieved and uniform bounds over classes of smooth test functions f are also discussed.