We consider the nonparametric regression estimation problem of recovering an unknown
response function f on the basis of spatially inhomogeneous data when the
design points follow a known density g with a finite number of
well-separated zeros. In particular, we consider two different cases: when
g has zeros of a polynomial order and when g has zeros
of an exponential order. These two cases correspond to moderate and severe data losses,
respectively. We obtain asymptotic (as the sample size increases) minimax lower bounds for
the L2-risk when f is assumed to belong to a
Besov ball, and construct adaptive wavelet thresholding estimators of f
that are asymptotically optimal (in the minimax sense) or near-optimal within a
logarithmic factor (in the case of a zero of a polynomial order), over a wide range of
Besov balls. The spatially inhomogeneous ill-posed problem that we investigate is
inherently more difficult than spatially homogeneous ill-posed problems like,
e.g., deconvolution. In particular, due to spatial irregularity,
assessment of asymptotic minimax global convergence rates is a much harder task than the
derivation of asymptotic minimax local convergence rates studied recently in the
literature. Furthermore, the resulting estimators exhibit very different behavior and
asymptotic minimax global convergence rates in comparison with the solution of spatially
homogeneous ill-posed problems. For example, unlike in the deconvolution problem, the
asymptotic minimax global convergence rates are greatly influenced not only by the extent
of data loss but also by the degree of spatial homogeneity of f.
Specifically, even if 1/g is non-integrable, one can recover
f as well as in the case of an equispaced design (in terms of
asymptotic minimax global convergence rates) when it is homogeneous enough since the
estimator is “borrowing strength” in the areas where f is adequately
sampled.