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Efficient estimation of multivariate analytic functions in cube-like domains. (English) Zbl 1132.62320
Summary: Let \(Q\) be an affine image of the unit cube in \(\mathbb{R}^d\). An unknown function \(f\) defined on \(Q\) is observed in the continuous heteroscedastic regression model \[ dV(x)= f(x)dx+ \varepsilon\sigma(x)dW(x),\qquad x\in Q. \] It is assumed that \(f\) admits an analytic continuation into a certain vicinity \(S_\gamma\) of \(Q\) in the \(d\)-dimensional complex space \(\mathbb{C}^d\). A special form of the noise variance \(\sigma^2(x)\) ensures that the information about \(f(x)\) provided by the data \(V(\cdot)\) is (nearly) independent of \(x\in\text{int\,}Q\). Projection-type estimates of \(f(x)\) are shown to be asymptotically efficient for any given \(x\in\text{int\,}Q\), as well as in all \(L_p(Q)\)-norms, \(1\leq p\leq\infty\), in both periodic and non-periodic cases.

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis