Penalized log-density estimation using Legendre polynomials. Issue 11 (1st November 2020)

• Record Type: Journal Article
• Authors: Lee, JungJun; Jhong, Jae-Hwan; Cho, Young-Rae; Kim, SungHwan; Koo, Ja-Yong
• Is Part Of: Communications in statistics. Volume 49:Issue 11(2020)
• Page Start: 2844
• Page End: 2860
• Publication Date: 2020-11-01
• Subjects: Penalized log-density estimation -- nonparametric density estimation -- Legendre polynomial basis -- coordinate descent algorithm -- ℓ1 penalty -- maximum tuning parameter; Mathematical statistics -- Periodicals; Mathematical statistics -- Data processing -- Periodicals; Digital computer simulation -- Periodicals; 519.5
• DOI: 10.1080/03610918.2018.1528360
• Languages: English
• ISSNs: 0361-0918

Abstract

Abstract: In this article, we present a penalized log-density estimation method using Legendre polynomials with ℓ 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 λ 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.