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A fast algorithm for univariate log‐concave density estimation
Author(s) -
Liu Yu,
Wang Yong
Publication year - 2018
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/anzs.12232
Subject(s) - mathematics , nonparametric statistics , univariate , quadratic growth , algorithm , extension (predicate logic) , maximum likelihood , function (biology) , convergence (economics) , newton's method , mathematical optimization , statistics , multivariate statistics , nonlinear system , computer science , physics , evolutionary biology , quantum mechanics , economics , biology , programming language , economic growth
Summary A new fast algorithm for computing the nonparametric maximum likelihood estimate of a univariate log‐concave density is proposed and studied. It is an extension of the constrained Newton method for nonparametric mixture estimation. In each iteration, the newly extended algorithm includes, if necessary, new knots that are located via a special directional derivative function. The algorithm renews the changes of slope at all knots via a quadratically convergent method and removes the knots at which the changes of slope become zero. Theoretically, the characterisation of the nonparametric maximum likelihood estimate is studied and the algorithm is guaranteed to converge to the unique maximum likelihood estimate. Numerical studies show that it outperforms other algorithms that are available in the literature. Applications to some real‐world financial data are also given.