Penalized log-density estimation using Legendre polynomials

Abstract

In this article, we present a penalized log-density estimation method using Legendre polynomials with ${\ell }_1$ penalty to adjust estimate’s smoothness. Re-expressing the logarithm of the density estimator via a linear combination of Legendre polynomials, we can estimate parameters by maximizing the penalized log-likelihood function. Besides, we proposed an implementation strategy that builds on the coordinate decent algorithm, together with the Bayesian information criterion (BIC). In particular, we derive a numerical solution to the maximum tuning parameter ${\lambda }_{\max }$ which leads to all zero coefficients and practically facilitates searching the optimal tuning parameter. Extensive simulation studies clearly show that our proposed estimator is computationally competitive with other existing nonparametric density estimators (e.g., kernel, kernel smooth and logspline estimators) benchmarked by the mean integrated squared errors (MISE) and the mean integrated absolute error (MIAE) under the experiment scenario of separated bimodal models in regard to the true density function. With an application to Old Faithful geyser data, our proposed method is found to effectively perform density estimation.

Keywords:

• Penalized log-density estimation
• nonparametric density estimation
• Legendre polynomial basis
• coordinate descent algorithm
• ${\ell }_1$ penalty
• maximum tuning parameter

Acknowledgments

We wish to thank two referees for providing valuable comments that improve quality of this paper.

Additional information

Funding

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2008619 and NRF2017R1C1B5017528).