Consistency of Lp-best monotone approximations

Under the hypothesis of the estimates being ‘well-behaved approximations’, the almost sure and sample- L p -consistency of the procedure is proved and as a main consequence the consistency of the L p best monotone approximation of the kernel regression estimate is obtained.

Abstract

The paper studies consistency properties of the empirical Lp-best monotone approximations of estimates of an unknown function. Under the hypothesis of the estimates being ‘well-behaved approximations’, we prove the almost sure and sample-Lp-consistency of the procedure. As a main consequence we obtain the consistency of the Lp-best monotone approximation of the kernel regression estimate. The obtention of this result involves as a previous fact of independent interest the empirical Lp-consistency of the kernel estimate of the regression function. The paper includes some simulations that illustrate the performance of the suggested method.