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

Karima Lagha and Smail Adjabi

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 12, 5941-5955

Abstract: We present in this work an orthogonal series density estimator based on random number of observations Nt. We give its statistical properties (bias, variance, mean square error, and mean square integrated error) and some asymptotic properties. We consider that Nt is independent of the observations and Nt→P∞$N_t \stackrel{P}{\rightarrow } \infty$ 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.

Date: 2017
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DOI: 10.1080/03610926.2015.1115075

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