How to cite this WIREs title:
WIREs Comp Stat

Computation of the nonparametric maximum likelihood estimate of a univariate log‐concave density

Focus Article

Yong Wang

Published Online: Aug 07 2018

DOI: 10.1002/wics.1452

Full article on Wiley Online Library: HTML PDF

Can't access this content? Tell your librarian.

Algorithms for computing the nonparametric maximum likelihood estimate of a univariate log‐concave density are briefly described, and some relevant issues discussed. The fast few algorithms are further numerically compared in a small‐scale simulation study. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC) Statistical and Graphical Methods of Data Analysis > Density Estimation Statistical and Graphical Methods of Data Analysis > Nonparametric Methods Algorithms and Computational Methods > Algorithms

Density, log‐density and gradient functions of the nonparametric maximum likelihood estimators (NPMLEs) obtained from generic samples (n = 1,000) drawn from Exponential(1) and Normal(0, 1), respectively. Knots are indicated by small circles

[ Normal View | Magnified View ]

References

Anderson‐Bergman,, C. (2014). R package ‘logconPH’: CoxPH model with log concave base‐line distribution. Retrieved from http://cran.r-project.org/package=logconPH
Anderson‐Bergman,, C., & Yu,, Y. (2016). Computing the log concave NPMLE for interval censored data. Statistics and Computing, 26, 813–826.
Ayer,, M., Brunk,, H. D., Ewing,, G. M., Reid,, W. T., & Silverman,, E. (1955). An empirical distribution function for sampling with incomplete information. Annals of Mathematical Statistics, 26, 641–647.
Balabdaoui,, F., Jankowski,, H., Rufibach,, K., & Pavlides,, M. (2013). Asymptotics of the discrete log‐concave maximum likelihood estimator and related applications. Journal of the Royal Statistical Society, Series B, 75, 769–790.
Best,, M. J., & Chakravarti,, N. (1990). Active set algorithms for isotonic regression ‐ a unifying framework. Mathematical Programming, 47, 425–439.
Cule,, M., Samworth,, R., & Stewart,, M. (2010). Maximum likelihood estimation of a multidimensional log‐concave density (with discussion). Journal of the Royal Statistical Society, Series B, 72, 545–576.
Dümbgen,, L., Freitag‐Wolf,, S., & Jongbloed,, G. (2006). Estimating a unimodal distribution from interval‐censored data. Journal of the American Statistical Association, 101, 1094–1106.
Dümbgen,, L., Hasler,, A., %26 Rufibach,, K. (2011). Active set and EM algorithms for log‐concave densities based on complete and censored data. (arXiv:0707.4643v4)
Dümbgen,, L., & Rufibach,, K. (2011). Logcondens: Computations related to univariate log‐concave density estimation. Journal of Statistical Software, 39, 1–28.
Fedorov,, V. V. (1972). Theory of optimal experiments. New York, NY: Academic Press.
Groeneboom,, P. (1991). Nonparametric maximum likelihood estimators for interval censoring and deconvolution (Technical Report No. 378). Department of Statistics, Stanford University.
Groeneboom,, P., Jongbloed,, G., & Wellner,, J. A. (2008). The support reduction algorithm for computing nonparametric function estimates in mixture models. Scandinavian Journal of Statistics, 35, 385–399.
Grotzinger,, S. J., & Witzgall,, C. (1984). Projection onto order simplexes. Applied Mathematics and Optimization, 12, 247–270.
Jongbloed,, G. (1998). The iterative convex minorant algorithm for nonparametric estimation. Journal of Computational and Graphical Statistics, 7, 301–321.
Kooperberg,, C., & Stone,, C. J. (1991). A study of logspline density estimation. Computational Statistics %26 Data Analysis, 12, 327–347.
Kooperberg,, C., & Stone,, C. J. (1992). Logspline density estimation for censored data. Journal of Computational and Graphical Statistics, 1, 301–328.
Lawson,, C. L., & Hanson,, R. J. (1974). Solving least squares problems. Englewood Cliffs, NJ: Prentice‐Hall, Inc.
Liu,, Y., & Wang,, Y. (2018a). A fast algorithm for univariate log‐concave density estimation. Australia %26 New Zealand Journal of Statistics, 60, 258–275.
Liu,, Y., %26 Wang,, Y. (2018b). R package cninlcd:Maximum likeli‐hood estimation of a log‐concave density function (version 1.2‐0). Retrieved from http : //cran . r‐pro j ect . org/package=cnmlcd
Nocedal,, J., & Wright,, S. (2006). Numerical optimization (2nd ed.). New York, NY: Springer.
Pardalos,, P. M., & Xue,, G. (1999). Algorithms for a class of isotonic regression problems. Algorithmica, 23, 211–222.
Rufibach,, K. (2007). Computing maximum likelihood estimators of a log‐concave density function. Journal of Statistical Computation and Simulation, 77, 561–574.
Rufibach,, K., %26 Dümbgen,, L. (2016). R package `logcondens`: Estimate a log‐concave probability density from iid observations (version 2.1.5). Retrieved from http://cran.r‐project.org/package=logcondens
Silverman,, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method. Annals of Statistics, 10, 795–810.
Terlaky,, T., & Vial,, J.‐P. (1998). Computing maximum likelihood estimators of convexdensity functions. SIAM Journal on Scientific Computing, 19, 675–694.
Wang,, Y. (2007). On fast computation of the non‐parametric maximum likelihood estimate of a mixing distribution. Journal of the Royal Statistical Society, Series B, 69, 185–198.
Wynn,, H. P. (1970). The sequential generation of D‐optimal experimental design. Annals of Mathematical Statistics, 41, 1655–1664.

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

Computation of the nonparametric maximum likelihood estimate of a univariate log‐concave density

Browse by Topic

Access to this WIREs title is by subscription only.

The latest WIREs articles in your inbox