We consider the nonparametric regression estimation problem of recovering an unknownresponse function f on the basis of spatially inhomogeneous data when thedesign points follow a known density g with a finite number ofwell-separated zeros. In particular, we consider two different cases: wheng has zeros of a polynomial order and when g has zerosof 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 forthe L2-risk when f is assumed to belong to aBesov ball, and construct adaptive wavelet thresholding estimators of fthat are asymptotically optimal (in the minimax sense) or near-optimal within alogarithmic factor (in the case of a zero of a polynomial order), over a wide range ofBesov balls. The spatially inhomogeneous ill-posed problem that we investigate isinherently 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 thederivation of asymptotic minimax local convergence rates studied recently in theliterature. Furthermore, the resulting estimators exhibit very different behavior andasymptotic minimax global convergence rates in comparison with the solution of spatiallyhomogeneous ill-posed problems. For example, unlike in the deconvolution problem, theasymptotic minimax global convergence rates are greatly influenced not only by the extentof data loss but also by the degree of spatial homogeneity of f.Specifically, even if 1/g is non-integrable, one can recoverf as well as in the case of an equispaced design (in terms ofasymptotic minimax global convergence rates) when it is homogeneous enough since theestimator is “borrowing strength” in the areas where f is adequatelysampled.

Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.

We consider a deconvolution problem of estimating a signal blurred with a random noise.The noise is assumed to be a stationary Gaussian process multiplied by a weight function function εh where h ∈ L2(R1) and ε is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax andBayes estimators. In the case of solutions having finite number ofderivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii, IMS Lecture Notes Monograph Series36 (2001) 419–433].