2022/05/20 by Vladimir Koltchinskii, Koltchinskii, Vladimir · 1 citation
Mathematics · #60B20} #62H25 #FOS: Mathematics #Mathematical Approximation and Integration #Primary: 62H12 #Secondary: 62G20 #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2205.10280
openalex publication_date 2022/05/20 · openalex created_date 2022/05/25 · openalex updated_date 2026/07/28
Let E be a separable Banach space and let X, X1,…, Xn, … 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 X1,…, Xn is naturally characterized by the so called effective rank of Σ: \bf r(Σ):= \frac\mathbb EΣ‖X‖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 X1,…, Xn. 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 \bf 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 \bf r(Σ)\lesssim nα 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=\mathbb Rd and for covariance operators Σ whose spectrum is bounded away from zero (in which case, \bf r(Σ)\asymp d).