## Nonparametric regression under qualitative smoothness assumptions.(English)Zbl 0737.62039

The author proposes a new nonparametric regression estimate. In contrast to the traditional approach of considering regression functions whose $$m$$-th derivates lie in a ball in the $$L_ \infty$$ or $$L_ 2$$ norm, he considers the class of functions whose $$(m-1)$$-st derivates consists of at most $$k$$ monotone pieces. For many applications this class seems more natural than the classical ones. The least squares estimator of this class is studied. It is shown that this estimator turns out to be a regression spline of order $$(m-1)$$, i.e. an $$(m-2)$$ times continuously differentiable function and piecewise a polynomial of degree $$(m-1)$$, with knot points depending on the observations.
It is shown that the speed of convergence of the estimator is as fast as in the classical case. Some remarks are made on possible algorithms for the computations of the estimator. Simulated data are used to compare the estimator with a kernel estimator.

### MSC:

 62G07 Density estimation
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