Georgia Institute of Technology College of Sciences

We will discuss a problem of estimation of functionals of the form $\tau_f(\Sigma):= {\rm tr} (f(\Sigma))$ of unknown covariance operator $\Sigma$ of a centered Gaussian random variable $X$ in a separable Hilbert space ${\mathbb H}$ based on i.i.d. observation $X_1,\dots, X_n$ of $X,$ where $f:{\mathbb R}\mapsto {\mathbb R}$ is a given function. A naive plug-in estimator $\tau_f(\hat \Sigma_n)$ based on the sample covariance operator $\hat \Sigma_n$ has a large bias and bias reduction methods are needed to construct estimators with better error rates. We develop estimators with reduced bias based on linear aggregation of several plug-in estimators with different sample sizes and obtain the error bounds for such estimators with explicit dependence on the sample size $n,$ the effective rank ${\bf r}(\Sigma)= \frac{tr(\Sigma)}{\|\Sigma\|}$ of covariance operator $\Sigma$ and the degree of smoothness of function $f.$