Total error in a plug-in estimator of level sets

Given a probability density f on R^d, the minimum volume set of probability content á can be estimated by the level set of the same probability content corresponding to a kernel estimator of f. We obtain convergence rates for this plug-in estimator with respect to a measure-based distance between sets. This distance has a straightforward interpretation in the context of cluster analysis

Local linear regression for functional predictor and scalar response

The aim of this work is to introduce a new nonparametric regression technique in the context of functional covariate and scalar response. We propose a local linear regression estimator and study its asymptotic behaviour. Its finite-sample performance is compared with a Nadayara-Watson type kernel regression estimator via a Monte Carlo study and the analysis of two real data sets. In all the scenarios considered, the local linear regression estimator performs better than the kernel one, in the sense that the mean squared prediction error and its standard deviation are lower

Image estimators based on marked bins

The problem of approximating an "image" S in R^d from a random sample of points is considered. If S is included in a grid of square bins, a plausible estimator of S is defined as the union of the "marked" bins (those containing a sample point). We obtain convergence rates for this estimator and study its performance in the approximation of the border of S. The estimation of "digitalized" images is also addressed by using a Vapnik-Chervonenkis approach. The practical aspects of implementation are discussed in some detail, including some technical improvements on the estimator, whose performance is checked through simulated as well as real data examples

Mean squared errors of small area estimators under a unit-level multivariate model

This work deals with estimating the vector of means of characteristics of small areas. In this context, a unit level multivariate model with correlated sampling errors is considered. An approximation is obtained for the mean squared and cross product errors of the empirical best linear unbiased predictors of the means. This approach has been implemented on a Monte Carlo study using economic data observed for a sample of Australian farms

Supervised classification for a family of Gaussian functional models

In the framework of supervised classification (discrimination) for functional data, it is shown that the optimal classification rule can be explicitly obtained for a class of Gaussian processes with "triangular" covariance functions. This explicit knowledge has two practical consequences. First, the consistency of the well-known nearest neighbors classifier (which is not guaranteed in the problems with functional data) is established for the indicated class of processes. Second, and more important, parametric and nonparametric plug-in classifiers can be obtained by estimating the unknown elements in the optimal rule. The performance of these new plug-in classifiers is checked, with positive results, through a simulation study and a real data example.Comment: 30 pages, 6 figures, 2 table

A test for convex dominance with respect to the exponential class based on an L1 distance

We consider the problem of testing if a non-negative random variable is dominated, in the convex order, by the exponential class. Under the null hypothesis, the variable is harmonic new better than used in expectation (HNBUE), a well-known class of ageing distributions in reliability theory. As a test statistic, we propose the L1 norm of a suitable distance between the empirical and the exponential distributions, and we completely determine its asymptotic properties. The practical performance of our proposal is illustrated with simulation studies, which show that the asymptotic test has a good behavior and power, even for small sample sizes. Finally, three real data sets are analyze

Parametric versus nonparametric tolerance regions indetection problems

A major problem in statistical quality control is to detect a change in the underlying distribution of independent sequentially observed random vectors. The case where the prechange distribution is Gaussian has been extensively analyzed. We are concerned here with the less usual non-normal multivariate case. The use of tolerance regions, defined in terms of density level sets, as detection tools arises as a natural choice in this general setup. The required level sets can be estimated in an obvious plug-in fashion, using either nonparametric or (when a parametric model is assumed) parametric density estimators. A result concerning the convergence rates of the error probabilities under a parametric model is obtained. Also, the performance of parametric and non-parametric methods is compared through a simulation study. Finally, a real data example is discussed. In general terms, we conclude that whereas the parametric estimates are, in theory, preferable when the corresponding model holds, the practical difficulties associated with their implementation make non-parametric methods a very reliable and flexible alternative