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We study a problem of estimation of smooth functionals of parameter $theta $ of Gaussian shift model $$ X=theta +xi, theta in E, $$ where $E$ is a separable Banach space and $X$ is an observation of unknown vector $theta$ in Gaussian noise $xi$ with zero mean and known covariance operator $Sigma.$ In particular, we develop estimators $T(X)$ of $f(theta)$ for functionals $f:Emapsto {mathbb R}$ of Holder smoothness $s>0$ such that $$ sup_{|theta|leq 1} {mathbb E}_{theta}(T(X)-f(theta))^2 lesssim Bigl(|Sigma| vee ({mathbb E}|xi|^2)^sBigr)wedge 1, $$ where $|Sigma|$ is the operator norm of $Sigma,$ and show that this mean squared error rate is minimax optimal at least in the case of standard Gaussian shift model ($E={mathbb R}^d$ equipped with the canonical Euclidean norm, $xi =sigma Z,$ $Zsim {mathcal N}(0;I_d)$). Moreover, we determine a sharp threshold on the smoothness $s$ of functional $f$ such that, for all $s$ above the threshold, $f(theta)$ can be estimated efficiently with a mean squared error rate of the order $|Sigma|$ in a small noise setting (that is, when ${mathbb E}|xi|^2$ is small). The construction of efficient estimators is crucially based on a bootstrap chain method of bias reduction. The results could be applied to a variety of special high-dimensional and infinite-dimensional Gaussian models (for vector, matrix and functional data).
Let $X^{(n)}$ be an observation sampled from a distribution $P_{theta}^{(n)}$ with an unknown parameter $theta,$ $theta$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation
Let $X$ be a centered Gaussian random variable in a separable Hilbert space ${mathbb H}$ with covariance operator $Sigma.$ We study a problem of estimation of a smooth functional of $Sigma$ based on a sample $X_1,dots ,X_n$ of $n$ independent observa
Let $X_1,dots, X_n$ be i.i.d. random variables sampled from a normal distribution $N(mu,Sigma)$ in ${mathbb R}^d$ with unknown parameter $theta=(mu,Sigma)in Theta:={mathbb R}^dtimes {mathcal C}_+^d,$ where ${mathcal C}_+^d$ is the cone of positively
We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_1,dots, X_n$ in a separable Hilbert space $mathbb{H}$ with unknown covariance operator $Sigma.$ The complexity of the problem is characterized by its effective
In prevalent cohort studies where subjects are recruited at a cross-section, the time to an event may be subject to length-biased sampling, with the observed data being either the forward recurrence time, or the backward recurrence time, or their sum