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

2024/02/17 by Vladimir Koltchinskii, Koltchinskii, Vladimir
Mathematics · #60B20 #62H25 #FOS: Mathematics #Primary: 62H12 #Statistical Methods and Inference #Statistics Theory (math.ST) #secondary: 62G20

paper · pdf · doi:10.48550/arxiv.2402.11321

openalex publication_date 2024/02/17 · openalex created_date 2024/02/22 · openalex updated_date 2026/07/28

Abstract

Let f:\mathbb R₊↦ \mathbb R be a smooth function with f(0)=0. A problem of estimation of a functional τ_f(Σ):= \rm tr(f(Σ)) of unknown covariance operator Σ in a separable Hilbert space \mathbb H based on i.i.d. mean zero Gaussian observations X₁,…, X_n with values in \mathbb H and covariance operator Σ is studied. Let Σ_n be the sample covariance operator based on observations X₁,…, X_n. Estimators T_f,m(X₁,…, X_n):= ∑_j=1^m C_j τ_f( Σ_nj) based on linear aggregation of several plug-in estimators τ_f( Σ_nj), where the sample sizes n/c≤ n₁<…

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