Let \(f:{\mathbb R}_+\mapsto {\mathbb R}\) be a smooth function with \(f(0)=0.\) A problem of estimation of a functional \(\tau _f(\Sigma ):= \mathrm {tr}(f(\Sigma ))\) of unknown covariance operator \(\Sigma \) 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 \(\Sigma \) is studied. Functionals \(\tau _f(\Sigma )\) for a sufficiently large class of test functions f define the spectral measure \(\mu _{\Sigma }\) of \(\Sigma \) by the following relationship: \(\tau _f(\Sigma )=\int _{{\mathbb R}_+}fd\mu _{\Sigma }.\) Let \(\hat \Sigma _n\) be the sample covariance operator based on observations \(X_1,\dots , X_n.\) Estimators \(\displaystyle \begin{aligned} T_{f,m}(X_1,\dots , X_n):= \sum _{j=1}^m C_j \tau _f(\hat \Sigma _{n_j}) \end{aligned} \) based on linear aggregation of several plug-in estimators \(\tau _f(\hat \Sigma _{n_j}),\) where the sample sizes \(n_1,\dots , n_m\) satisfy the condition \(n/c\leq n_1<\dots <n_m_leq n_="" for="" some="" _c="">1\) and coefficients \(C_1,\dots , C_m\) are chosen to reduce the bias, are considered. Estimator \(T_{f,m}(X_1,\dots , X_n)\) could be also represented as an integral \(\int _{{\mathbb R}_+}fd\hat \mu _{n,m},\) where \(\hat \mu _{n,m} := \sum _{j=1}^m C_j \mu _{\hat \Sigma _{n_j}}\) is a signed measure in \({\mathbb R}_+\) providing an estimator of the spectral measure \(\mu _{\Sigma }.\) The complexity of the problem is characterized by the effective rank \(\mathbf {r}(\Sigma ):= \frac {\mathrm {tr}(\Sigma )}{\|\Sigma \|}\) of covariance operator \(\Sigma .\) It is shown that, if \(f\in C^{m+1}({\mathbb R}_+)\) for some \(m\geq 2,\) \(\|f''\|_{L_{\infty }}\lesssim 1,\) \(\|f^{(m+1)}\|_{L_{\infty }}\lesssim 1\) , \(\|\Sigma \|\ \lesssim 1\) , and \(\mathbf {r}(\Sigma )\lesssim n,\) then \(\displaystyle \begin{aligned} \|\hat T_{f,m}(X_1,\dots , X_n)-\tau _f(\Sigma )\|_{L_2} \lesssim _m \frac {\|\Sigma f'(\Sigma )\|_2}{\sqrt {n}} + \frac {\mathbf {r}(\Sigma )}{n}+ \mathbf {r}(\Sigma )\Big (\sqrt {\frac {\mathbf {r}(\Sigma )}{n}}\Big )^{m+1}. \end{aligned} \) 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 is discussed. A symmetrized (jackknife) version \(\check T_{f,m}(X_1,\dots , X_n)=\int _{{\mathbb R}_+}fd\check \mu _{n,m}\) of estimator \(\hat T_{f,m}(X_1,\dots , X_n)\) is also considered and, for this estimator, normal approximation bounds and asymptotic efficiency have been proved. Finally, bounds on the sup-norms of stochastic processes \(n^{1/2}\int _{{\mathbb R}_+}fd(\hat \mu _{n,m}-\mu _{\Sigma }), f\in {\mathcal F}\) and \(n^{1/2}\int _{{\mathbb R}_+}fd(\check \mu _{n,m}-\mu _{\Sigma }), f\in {\mathcal F}\) for classes \({\mathcal F}\) of smooth functions f as well as the results on Gaussian approximation of the second process are also discussed.</n_m_leq>

Estimation of Trace Functionals and Spectral Measures of Covariance Operators in Gaussian Models

会议论文
发表日期：

2026年

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Vladimir Koltchinskii

摘要

Let \(f:{\mathbb R}_+\mapsto {\mathbb R}\) be a smooth function with \(f(0)=0.\) A problem of estimation of a functional \(\tau _f(\Sigma ):= \mathrm {tr}(f(\Sigma ))\) of unknown covariance operator \(\Sigma \) 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 \(\Sigma \) is studied. Functionals \(\tau _f(\Sigma )\) for a sufficiently large class of test functions f define the spectral measure \(\mu _{\Sigma }\) of \(\Sigma \) by the following relationship: \(\tau _f(\Sigma )=\int _{{\mathbb R}_+}fd\mu _{\Sigma }.\) Let \(\hat \Sigma _n\) be the sample covariance operator based on observations \(X_1,\dots , X_n.\) Estimators \(\displaystyle \begin{aligned} T_{f,m}(X_1,\dots , X_n):= \sum _{j=1}^m C_j \tau _f(\hat \Sigma _{n_j}) \end{aligned} \) based on linear aggregation of several plug-in estimators \(\tau _f(\hat \Sigma _{n_j}),\) where the sample sizes \(n_1,\dots , n_m\) satisfy the condition \(n/c\leq n_1<\dots <n_m_leq n_="" for="" some="" _c="">1\) and coefficients \(C_1,\dots , C_m\) are chosen to reduce the bias, are considered. Estimator \(T_{f,m}(X_1,\dots , X_n)\) could be also represented as an integral \(\int _{{\mathbb R}_+}fd\hat \mu _{n,m},\) where \(\hat \mu _{n,m} := \sum _{j=1}^m C_j \mu _{\hat \Sigma _{n_j}}\) is a signed measure in \({\mathbb R}_+\) providing an estimator of the spectral measure \(\mu _{\Sigma }.\) The complexity of the problem is characterized by the effective rank \(\mathbf {r}(\Sigma ):= \frac {\mathrm {tr}(\Sigma )}{\|\Sigma \|}\) of covariance operator \(\Sigma .\) It is shown that, if \(f\in C^{m+1}({\mathbb R}_+)\) for some \(m\geq 2,\) \(\|f''\|_{L_{\infty }}\lesssim 1,\) \(\|f^{(m+1)}\|_{L_{\infty }}\lesssim 1\) , \(\|\Sigma \|\ \lesssim 1\) , and \(\mathbf {r}(\Sigma )\lesssim n,\) then \(\displaystyle \begin{aligned} \|\hat T_{f,m}(X_1,\dots , X_n)-\tau _f(\Sigma )\|_{L_2} \lesssim _m \frac {\|\Sigma f'(\Sigma )\|_2}{\sqrt {n}} + \frac {\mathbf {r}(\Sigma )}{n}+ \mathbf {r}(\Sigma )\Big (\sqrt {\frac {\mathbf {r}(\Sigma )}{n}}\Big )^{m+1}. \end{aligned} \) 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 is discussed. A symmetrized (jackknife) version \(\check T_{f,m}(X_1,\dots , X_n)=\int _{{\mathbb R}_+}fd\check \mu _{n,m}\) of estimator \(\hat T_{f,m}(X_1,\dots , X_n)\) is also considered and, for this estimator, normal approximation bounds and asymptotic efficiency have been proved. Finally, bounds on the sup-norms of stochastic processes \(n^{1/2}\int _{{\mathbb R}_+}fd(\hat \mu _{n,m}-\mu _{\Sigma }), f\in {\mathcal F}\) and \(n^{1/2}\int _{{\mathbb R}_+}fd(\check \mu _{n,m}-\mu _{\Sigma }), f\in {\mathcal F}\) for classes \({\mathcal F}\) of smooth functions f as well as the results on Gaussian approximation of the second process are also discussed.