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

Vladimir Koltchinskii

Abstract

Let $f:{\mathbb R}_+\mapsto {\mathbb R}$ 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 $X_1,\dots, X_n$ with values in ${\mathbb H}$ and covariance operator $Σ$ is studied. Let $\hat Σ_n$ be the sample covariance operator based on observations $X_1,\dots, X_n.$ Estimators \begin{align*} T_{f,m}(X_1,\dots, X_n):= \sum_{j=1}^m C_j τ_f(\hat Σ_{n_j}) \end{align*} based on linear aggregation of several plug-in estimators $τ_f(\hat Σ_{n_j}),$ where the sample sizes $n/c\leq n_1<\dots<n_m\leq n$ and coefficients $C_1,\dots, C_n$ are chosen to reduce the bias, are considered. The complexity of the problem is characterized by the effective rank ${\bf r}(Σ):= \frac{{\rm tr}(Σ)}{\|Σ\|}$ of covariance operator $Σ.$ It is shown that, if $f\in C^{m+1}({\mathbb R}_+)$ for some $m\geq 2,$ $\|f''\|_{L_{\infty}}\lesssim 1,$ $\|f^{(m+1)}\|_{L_{\infty}}\lesssim 1,$ $\|Σ\|\lesssim 1$ and ${\bf r}(Σ)\lesssim n,$ then \begin{align*} & \|\hat T_{f,m}(X_1,\dots, X_n)-τ_f(Σ)\|_{L_2} \lesssim_m \frac{\|Σf'(Σ)\|_2}{\sqrt{n}} + \frac{{\bf r}(Σ)}{n}+ {\bf r}(Σ)\Bigl(\sqrt{\frac{{\bf r}(Σ)}{n}}\Bigr)^{m+1}. \end{align*} Similar bounds have been proved for the $L_{p}$-errors and some other Orlicz norm errors of estimator $\hat T_{f,m}(X_1,\dots, X_n).$ 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.

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

Abstract

Let be a smooth function with A problem of estimation of a functional of unknown covariance operator in a separable Hilbert space based on i.i.d. mean zero Gaussian observations with values in and covariance operator is studied. Let be the sample covariance operator based on observations Estimators \begin{align*} T_{f,m}(X_1,\dots, X_n):= \sum_{j=1}^m C_j τ_f(\hat Σ_{n_j}) \end{align*} based on linear aggregation of several plug-in estimators where the sample sizes and coefficients are chosen to reduce the bias, are considered. The complexity of the problem is characterized by the effective rank of covariance operator It is shown that, if for some and then \begin{align*} & \|\hat T_{f,m}(X_1,\dots, X_n)-τ_f(Σ)\|_{L_2} \lesssim_m \frac{\|Σf'(Σ)\|_2}{\sqrt{n}} + \frac{{\bf r}(Σ)}{n}+ {\bf r}(Σ)\Bigl(\sqrt{\frac{{\bf r}(Σ)}{n}}\Bigr)^{m+1}. \end{align*} Similar bounds have been proved for the -errors and some other Orlicz norm errors of estimator The optimality of these error rates, other estimators for which asymptotic efficiency is achieved and uniform bounds over classes of smooth test functions are also discussed.
Paper Structure (7 sections, 43 theorems, 382 equations)

This paper contains 7 sections, 43 theorems, 382 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Concentration and normal approximation bounds
  4. Bounds on sup-norms of empirical spectral processes
  5. Bounding the bias of estimator $\hat{T}_f(X_1,\dots, X_n)$
  6. Proofs of the main results
  7. Proofs of lower bounds and related results

Key Result

Proposition 2.1

Let $f\in C^1({\hbox{R}}_+)$ with $f(0)=0$ and $\|f'\|_{\rm Lip}<\infty.$ Then, for all $p\geq 1,$ If, in addition, $\|f'\|_{L_{\infty}}<\infty,$ this implies that and, moreover,

Theorems & Definitions (77)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 67 more