Abstract
The Propagation-Separation Approach is an iterative procedure for pointwise estimation of local constant and local polynomial functions. The estimator is defined as a weighted mean of the observations with data-driven, iteratively updated weights. Within homogeneous regions it ensures a similar behavior as non-adaptive smoothing (propagation), while avoiding smoothing among distinct regions (separation). In order to enable a proof of stability of estimates, the authors of the original study introduced an additional memory step aggregating the estimators of the successive iteration steps. Here, we study theoretical properties of the simplified algorithm, where the memory step is omitted. In particular, we introduce a new strategy for the choice of the adaptation parameter yielding propagation and stability for local constant functions with sharp discontinuities.
Highlights
The estimator is defined as a weighted mean of the observations with data-driven, iteratively updated weights. Within homogeneous regions it ensures a similar behavior as non-adaptive smoothing, while avoiding smoothing among distinct regions
We show for a homogeneous setting that the propagation condition yields with Equation (A.1) in Appendix A an exponential bound for P(NKL(θi(k), θ) > z), the excess probability of the Kullback-Leibler divergence between the adaptive estimator θi(k) and the true parameter θ
This study provides theoretical properties for a simplified version of the Propagation-Separation Approach, where the memory step is omitted from the algorithm
Summary
We show for piecewise constant functions that the adaptivity of the method yields similar results even if the memory step is removed from the algorithm. This gains importance as it turned out, that for practical use the memory step is questionable. We aim to justify the omittance of the memory step, but we do not compare the results with other estimation methods or evaluate the estimation error since this has been done in previous works by Polzehl and Spokoiny, see [14, 15, 16]. We compare the theoretical results and discuss the impact of the memory step for the case that the unknown parameter function complies with the local constant model. Consequences of model misspecification will be analyzed in a separate study
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