Non parametric sequential estimation of the probability density function by orthogonal series

We present in this work an orthogonal series density estimator based on random number of observations N-t. We give its statistical properties (bias, variance, mean square error, and mean square integrated error) and some asymptotic properties. We consider that N-t is independent of the observations and as t . We show that Kronmal-Tarter method for choosing the smoothing parameter is still valid in the case where the sample size is random. A detailed study with cosine basis is presented. The estimator obtained is a probability density, asymptotically unbiased and consistent. A simulation is used in order to study the behavior of the density estimator and shows that the estimator is performant.