## 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.

### MSC:

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