Optimal designs for nonparametric estimation of zeros of regression functions (Q2913243)

In a nonparametric model with independent errors: \(y_i=m\left (x_i\right)+\varepsilon_i\), \(i=1,\dots ,n\), \(x_i\in \mathbb {R}\), the regression function \(m(\cdot)\) or its \(\nu\)th derivative \(m^{\left (\nu \right)}(\cdot)\), are estimated by the Gasser-Müller kernel estimators. The optimization of the bandwidths of the kernels is also considered. Zeros, i.e., solutions of \(m\left (x\right)=0\), or of \(m^{\left (\nu \right)}\left (x\right)=0\) are then estimated by zeros of the corresponding kernel estimators. The design \(x_1,\dots ,x_n\) is given by the \(k/n\)-quantiles of a density \(f\left (x\right)\), which is to be chosen optimally with regard to the asymptotic integrated variance of estimators of zeros: \(\int {\operatorname {Var}}\bigl (\hat {\xi}_{n,b_n,\nu}\mid \xi _\nu =u\bigr) a\left (u\right) du\), where \(\xi _\nu \) and \(\hat {\xi}_{n,b_n,\nu}\) are zeros of \(m^{\left (\nu \right)}(\cdot)\) and of its estimator, and \(a\left (u\right)\) is a supposed prior density of zeros of \(m^{\left (\nu \right)}(\cdot)\). The explicit form of the optimal \(f\left (x\right)\) is then presented. For example, in the case of estimating zeros of \(m(\cdot)\) with constant bandwidths it has a simple form \(f\left (x\right)=\bigl \{w\left (x\right) a\left (x\right) \bigr \}^{1/2}\), where \(w\left (x_i\right)\) is proportional to the variance of \(\varepsilon _i\). A simulation study is added leading to a conclusion that the choice of an appropriate bandwidth is more influential than the choice of an optimal design. The set-up is clearly written, the results are in a condensed form. A valuable survey of references on this nonstandard design problem is presented as well.