A note on maximum likelihood density estimation using a proxy of the Kullback-Leibler distance.

Given a random sample from a continuous and positive density f , the logistic transformation is applied and a log density estimate is provided by using B-splines. The log density estimate maximizes the likelihood function which has equivalent solution when subject to a constraint that guarantees identifiability of the model. An finite approximation is provided and the number of basis functions which acts as the smoothing parameter is estimated by minimizing a proxy of the Kullback-Leibler distance. Keywords: non-parametric density estimation; B-splines; partitions of unity. 1 Introduction Let X 1 ; : : : ; X n be i.i.d. random variables with an unknown continuous and po- Postal address: Departamento de Estat'istica, IMECC, Cidade Universit'aria "Zeferino Vaz", Caixa Postal 6065, 13.081-970 - Campinas, SP - BRAZIL, e-mail address: dias@ime.unicamp.br 1 sitive probability density function f on a compact interval X . The goal of this paper is to estimate the density f from the d..