Some stochastics on monotone functions
We are concerned with monotone real functions in two different directions. However in both cases we pay special attention to obtain almost sure convergence results and the problem is related with measuring departures of monotonicity. Very often in mathematics the measurement of some kind of precision or anomaly is based on the consideration of metrics related to the problem under study. The different \(L_r\)- metrics possess a wide range of properties which make them specially suitable in these situations: typically it is possible to get a good approximation for the study of the desired problem by considering some suitable \(L_r\)-metric. In probability and statistics, this leads in a natural fashion to the usual fact of making reference to optimal properties formulated in terms of \(L_r\)-approximations. There is a vast literature about the advantages of considering any one of the \(L_r\)-norms. Therefore, in spite of the existence of some other possibilities to measure departures from the isotonicity (for instance one might use the Hausdorff metric), we suggest to measure departures of isotonicity, with respect to the different \(L_r\)-metrics, by considering the distance to the class of nondecreasing functions. This is carried out through the introduction of the \(L_r\)-DIP, \(1\leq r\leq\infty\), in Section 2. The other direction of our study arises in the nonparametric regression context, where the knowledge of the function is limited to a random sample relating the joint distribution of a pair of random variables. Therefore the estimation of the \(L_r\)-DIP must be based on a preliminary estimation of that function. We study this problem by considering the \(L_r\)-DIP measured over the estimation of the regression. When such an estimation is sampling \(L_r\)-consistent, we obtain strong consistency of estimators of the theoretical \(L_r\)-DIP. We particularize our study to the well-known Nadaraya-Watson estimator, obtaining strong consistency under the usual hypotheses in the literature to get \(L_r\)-consistency of the kernel estimator. In Section 3 we provide the technical support for both estimation problems. We prove the strong consistency of the sample version of the \(L_r\)-DIP. This section also includes the proof of the consistency of the corresponding best monotone approximations. Finally, in the Appendix we show some results related with the quantile function and Skorokhod's a.s. representation theorem.
scientific article; zbMATH DE number 758525
scientific article