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Probability density estimation using spline

Tatsuhei Iwasaki
Ecole Polytechnique, Paris

Stage de D.E.A

Octobre 1999


Of the methods used to estimate probability densities of unknown functional form, one of the most used is the histogram. However it is discontinuous. Clearly the histogram should offer interesting possibilities for generalization. The histogram is a nonparametric estimator in the sense that the number of characterizing parameters increases without limit as the sample size goes to infinity. People mainly studied kernel estimate, first introduced by Parzen and Rosenbatt. This estimate is nothing but an equiprobable mixture of n similar- shaped densities (kernels) centered at the data point. We present here one approach to the nonparametric probability density estimation using splines. Our approach can be considered as a generalization of fi xed grid histogram. The simplest way to generalize the fixed grid histogram is to use different bases than step functions. We mainly study the splines which are used intensively in signal processing for they present many attractive mathematical and computational features. We show also numerical examples using theoretical results obtained in Section II.