Exponential rates for kernel density estimation under association

Carla Henriques and Paulo Eduardo Oliveira

Statistica Neerlandica, 2005, vol. 59, issue 4, 448-466

Abstract: The estimation of density based on positive dependent samples has been studied recently with consistency and asymptotic normality results being obtained. In with regard to the characterization on decrease rates the results have been scarce. We prove two versions of an exponential inequality: one assuming stationarity and association alone and the other under a further assumption on the joint distributions of the sample. These inequalities are then used to prove exponential decrease rates for the kernel estimator of the density with a uniform version over compact sets. The conditions assumed impose convenient decrease rates on the covariance structure of the sample. Some examples supposing geometrical or polynomial decrease rates on the covariances that fulfill our assumptions are presented in the last section. Explicit almost sure rates are derived for geometrically decreasing covariances. Under the extra assumption on the joint distributions the rates are close to the best known ones for independent variables.

Date: 2005
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