Nonparametric estimation of directional highest density regions

Highest density regions (HDRs) are defined as level sets containing sample points of relatively high density. Although Euclidean HDR estimation from a random sample, generated from the underlying density, has been widely considered in the statistical literature, this problem has not been contemplated for directional data yet. In this work, directional HDRs are formally defined and plug-in estimators based on kernel smoothing and associated confidence regions are proposed. We also provide a new suitable bootstrap bandwidth selector for plug-in HDRs estimation based on the minimization of an error criteria that involves the Hausdorff distance between the boundaries of the theoretical and estimated HDRs. An extensive simulation study shows the performance of the resulting estimator for the circle and for the sphere. The methodology is applied to analyze two real data sets in animal orientation and seismology.

Bagging cross-validated bandwidths with application to Big Data

Hall & Robinson (2009) proposed and analyzed the use of bagged cross-validation to choose the bandwidth of a kernel density estimator. They established that bagging greatly reduces the noise inherent in ordinary cross-validation, and hence leads to a more efficient bandwidth selector. The asymptotic theory of Hall & Robinson (2009) assumes that N, the number of bagged subsamples, is ∞. We expand upon their theoretical results by allowing N to be finite, as it is in practice. Our results indicate an important difference in the rate of convergence of the bagged cross-validation bandwidth for the cases N = ∞ and N < ∞. Simulations quantify the improvement in statistical efficiency and computational speed that can result from using bagged cross-validation as opposed to a binned implementation of ordinary crossvalidation. The performance of the bagged bandwidth is also illustrated on a real, very large, data set. Finally, a byproduct of our study is the correction of errors appearing in the Hall & Robinson (2009) expression for the asymptotic mean squared error of the bagging selector.