dc.contributor.author Schick, Anton dc.contributor.author Wefelmeyer, Wolfgang dc.date.accessioned 2008-03-20T12:00:16Z dc.date.accessioned 2016-10-05T14:14:07Z dc.date.available 2008-03-20T12:00:16Z dc.date.available 2016-10-05T14:14:07Z dc.date.issued 2008-03-11 dc.identifier.uri http://publications.mfo.de/handle/mfo/1125 dc.description Research in Pairs 2007 en_US dc.description.abstract It has been shown recently that, under an appropriate integrability condition, densities of functions of independent and identically distributed random variables can be estimated at the parametric rate by a local U-statistic, and a functional central limit theorem holds. For the sum of two squared random variables, the integrability condition typically fails. We show that then the estimator behaves differently for different arguments. At points in the support of the squared random variable, the rate of the estimator slows down by a logarithmic factor and is independent of the bandwidth, but the asymptotic variance depends on the rate of the bandwidth, and otherwise only on the density of the squared random variable at this point and at zero. A functional central limit theorem cannot hold. Of course, for bounded random variables, the sum of squares is more spread out than a single square. At points outside the support of the squared random variable, the estimator behaves classically. Now the rate is again parametric, the asymptotic variance has a different form and does not depend on the bandwidth, and a functional central limit theorem holds. en_US dc.language.iso en en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2008,07 dc.subject convolution density estimator en_US dc.subject smoothness of convolutions en_US dc.subject asymptotically linear estimator en_US dc.title Non-Standard Behavior of Density Estimators for Sums of Squared Observations en_US dc.type Preprint en_US dc.identifier.doi 10.14760/OWP-2008-07 local.scientificprogram Research in Pairs 2007 local.series.id OWP-2008-07 local.subject.msc 62
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