SUMMARYThis paper analyzes outlier detection for functional data by means of functional depths, which measures the centrality of a given curve within a group of trajectories providing center-outward orderings of the set of curves. We give some insights of the usefulness of looking for outliers in functional datasets and propose a method based in depths for the functional outlier detection. The performance of the proposed procedure is analyzed by several Monte Carlo experiments. Finally, we illustrate the procedure by finding outliers in a dataset of NO x (nitrogen oxides) emissions taken from a control station near an industrial area.

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Concerning the estimation of linear parameters in small areas, a nested-error regression model is assumed for the values of the target variable in the units of a finite population. Then, a bootstrap procedure is proposed for estimating the mean squared error (MSE) of the EBLUP under the finite population setup. The consistency of the bootstrap procedure is studied, and a simulation experiment is carried out in order to compare the performance of two different bootstrap estimators with the approximation given by Prasad and Rao [Prasad, N.G.N. and Rao, J.N.K., 1990, The estimation of the mean squared error of small-area estimators. Journal of the American Statistical Association, 85, 163-171.]. In the numerical results, one of the bootstrap estimators shows a better bias behavior than the Prasad-Rao approximation for some of the small areas and not much worse in any case. Further, it shows less MSE in situations of moderate heteroscedasticity and under mispecification of the error distribution as normal when the true distribution is logistic or Gumbel. The proposed bootstrap method can be applied to more general types of parameters (linear of not) and predictors.

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