2017/10/25 by Koltchinskii, Vladimir · 1 citation
#62H12 #FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1710.09072
Let X be a centered Gaussian random variable in a separable Hilbert space \mathbb H with covariance operator Σ. We study a problem of estimation of a smooth functional of Σ based on a sample X1,… ,Xn of n independent observations of X. More specifically, we are interested in functionals of the form ⟨ f(Σ), B⟩, where f:\mathbb R↦ \mathbb R is a smooth function and B is a nuclear operator in \mathbb H. We prove concentration and normal approximation bounds for plug-in estimator ⟨ f( Σ),B⟩, Σ:=n-1∑j=1n Xj⊗ Xj being the sample covariance based on X1,…, Xn. These bounds show that ⟨ f( Σ),B⟩ is an asymptotically normal estimator of its expectation \mathbb EΣ ⟨ f( Σ),B⟩ (rather than of parameter of interest ⟨ f(Σ),B⟩) with a parametric convergence rate O(n-1/2) provided that the effective rank \bf r(Σ):= \frac\bf tr(Σ)‖Σ‖ (\rm tr(Σ) being the trace and ‖Σ‖ being the operator norm of Σ) satisfies the assumption \bf r(Σ)=o(n). At the same time, we show that the bias of this estimator is typically as large as \frac\bf r(Σ)n (which is larger than n-1/2 if \bf r(Σ)≥ n1/2). In the case when \mathbb H is finite-dimensional space of dimension d=o(n), we develop a method of bias reduction and construct an estimator ⟨ h( Σ),B⟩ of ⟨ f(Σ),B⟩ that is asymptotically normal with convergence rate O(n-1/2). Moreover, we study asymptotic properties of the risk of this estimator and prove minimax lower bounds for arbitrary estimators showing the asymptotic efficiency of ⟨ h( Σ),B⟩ in a semi-parametric sense.