Electronic Journal of Statistics

Univariate log-concave density estimation with symmetry or modal constraints

Abstract

We study nonparametric maximum likelihood estimation of a log-concave density function $f_{0}$ which is known to satisfy further constraints, where either (a) the mode $m$ of $f_{0}$ is known, or (b) $f_{0}$ is known to be symmetric about a fixed point $m$. We develop asymptotic theory for both constrained log-concave maximum likelihood estimators (MLE’s), including consistency, global rates of convergence, and local limit distribution theory. In both cases, we find the MLE’s pointwise limit distribution at $m$ (either the known mode or the known center of symmetry) and at a point $x_{0}\ne m$. Software to compute the constrained estimators is available in the R package logcondens.mode.

The symmetry-constrained MLE is particularly useful in contexts of location estimation. The mode-constrained MLE is useful for mode-regression. The mode-constrained MLE can also be used to form a likelihood ratio test for the location of the mode of $f_{0}$. These problems are studied in separate papers. In particular, in a separate paper we show that, under a curvature assumption, the likelihood ratio statistic for the location of the mode can be used for hypothesis tests or confidence intervals that do not depend on either tuning parameters or nuisance parameters.

Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 2391-2461.

Dates
First available in Project Euclid: 23 July 2019

https://projecteuclid.org/euclid.ejs/1563868824

Digital Object Identifier
doi:10.1214/19-EJS1574

Mathematical Reviews number (MathSciNet)
MR3983344

Zentralblatt MATH identifier
1422.62137

Subjects
Primary: 62G07: Density estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Citation

Doss, Charles R.; Wellner, Jon A. Univariate log-concave density estimation with symmetry or modal constraints. Electron. J. Statist. 13 (2019), no. 2, 2391--2461. doi:10.1214/19-EJS1574. https://projecteuclid.org/euclid.ejs/1563868824

References

• Azadbakhsh, M., Jankowski, H. and Gao, X. (2014). Computing confidence intervals for log-concave densities., Comput. Statist. Data Anal. 75 248–264.
• Balabdaoui, F. and Doss, C. R. (2018). Inference for a two-component mixture of symmetric distributions under log-concavity., Bernoulli 24 1053–1071.
• Balabdaoui, F., Rufibach, K. and Wellner, J. A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density., Ann. Statist. 37 1299–1331.
• Balabdaoui, F. and Wellner, J. A. (2007). Estimation of a $k$-monotone density: limit distribution theory and the spline connection., Ann. Statist. 35 2536–2564.
• Billingsley, P. (1999)., Convergence of Probability Measures, second ed. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York. A Wiley-Interscience Publication.
• Chacón, J. E. (2018a). Mixture model modal clustering., Advances in Data Analysis and Classification 12 1–26.
• Chacón, J. E. (2018b). The Modal Age of Statistics., arXiv:1807.02789.
• Chang, G. T. and Walther, G. (2007). Clustering with mixtures of log-concave distributions., Comput. Statist. Data Anal. 51 6242–6251.
• Chen, Y.-C., Genovese, C. R. and Wasserman, L. (2015). Asymptotic theory for density ridges., Ann. Statist. 43 1896–1928.
• Chen, Y.-C., Genovese, C. R., Tibshirani, R. J. and Wasserman, L. (2016). Nonparametric modal regression., Ann. Statist. 44 489–514.
• Cule, M. and Samworth, R. (2010). Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density., Electron. J. Stat. 4 254–270.
• Cule, M., Samworth, R. and Stewart, M. (2010). Maximum likelihood estimation of a multi-dimensional log-concave density., J. R. Stat. Soc. Ser. B Stat. Methodol. 72 545–607.
• Dieudonné, J. (1969)., Foundations of Modern Analysis. Academic Press, New York-London.
• Doss, C. R. (2013a). logcondens.mode: Compute MLE of Log-Concave Density on R with Fixed Mode, and Perform Inference for the Mode. R package version 1.0.1. Available from, http://CRAN.R-project.org/package=logcondens.mode.
• Doss, C. R. (2013b). Shape-constrained inference for concave-transformed densities and their modes PhD thesis, University of, Washington.
• Doss, C. R. and Wellner, J. A. (2016). Global rates of convergence of the MLEs of log-concave and $s$-concave densities., Ann. Statist. 44 954–981. With supplementary material available online.
• Doss, C. R. and Wellner, J. A. (2019). Inference for the mode of a log-concave density., Annals of Statistics 47 to appear. arXiv:1611.10348.
• Dümbgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log-concave density and its distribution function: basic properties and uniform consistency., Bernoulli 15 40–68.
• Dümbgen, L. and Rufibach, K. (2011). logcondens: computations related to univariate log-concave density estimation., Journal of Statistical Software 39 1–28.
• Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression., Ann. Statist. 39 702–730.
• Eilers, P. H. C. and Borgdorff, M. W. (2007). Non-parametric log-concave mixtures., Comput. Statist. Data Anal. 51 5444–5451.
• Folland, G. B. (1999)., Real Analysis, second ed. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York. Modern techniques and their applications, A Wiley-Interscience Publication.
• Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). A canonical process for estimation of convex functions: the “invelope” of integrated Brownian motion $+t^4$., Ann. Statist. 29 1620–1652.
• Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Estimation of a convex function: characterizations and asymptotic theory., Ann. Statist. 29 1653–1698.
• Hall, P. (1984). On unimodality and rates of convergence for stable laws., J. London Math. Soc. (2) 30 371–384.
• Han, Q. and Wellner, J. A. (2016). Approximation and estimation of $s$-concave densities via Rényi divergences., Ann. Statist. 44 1332–1359.
• Kim, J. and Pollard, D. (1990). Cube root asymptotics., Ann. Statist. 18 191–219.
• Kim, A. K. H. and Samworth, R. J. (2016). Global rates of convergence in log-concave density estimation., Ann. Statist. 44 2756–2779.
• Lachal, A. (1997). Local asymptotic classes for the successive primitives of Brownian motion., Ann. Probab. 25 1712–1734.
• Laha, N. (2019). Some semiparametric models under log-concavity. PhD thesis in, progress.
• Mason, D. M. and van Zwet, W. R. (1987). A refinement of the KMT inequality for the uniform empirical process., Ann. Probab. 15 871–884.
• Pal, J. K., Woodroofe, M. and Meyer, M. (2007). Estimating a Polya frequency function$_2$. In, Complex datasets and inverse problems. IMS Lecture Notes Monogr. Ser. 54 239–249. Inst. Math. Statist., Beachwood, OH.
• Polonik, W. (1995). Measuring mass concentrations and estimating density contour clusters—an excess mass approach., Ann. Statist. 23 855–881.
• Pu, X. and Arias-Castro, E. (2017). Semiparametric estimation of symmetric mixture models with monotone and log-concave densities., arXiv:1702.08897.
• Qiao, W. and Polonik, W. (2016). Theoretical analysis of nonparametric filament estimation., Ann. Statist. 44 1269–1297.
• R Core Team, (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
• Rockafellar, R. T. (1970)., Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J.
• Romano, J. P. (1987). On bootstrapping the joint distribution of the location and size of the mode PhD thesis, University of California, Berkeley.
• Royden, H. L. (1988)., Real Analysis, Third ed. Macmillan Publishing Company, New York.
• Rufibach, K. (2006). Log-concave density estimation and bump hunting for IID observations. PhD thesis, Univ. Bern and, Gottingen.
• Seregin, A. and Wellner, J. A. (2010). Nonparametric estimation of multivariate convex-transformed densities., Ann. Statist. 38 3751–3781. With supplementary material available online.
• Shorack, G. R. (2000)., Probability for Statisticians. Springer Texts in Statistics. Springer-Verlag, New York.
• Shorack, G. R. and Wellner, J. A. (2009)., Empirical Processes with Applications to Statistics. Classics in Applied Mathematics 59. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
• Silverman, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method., Ann. Statist. 10 795–810.
• Sinai, Y. G. (1992). Statistics of shocks in solutions of inviscid Burgers equation., Comm. Math. Phys. 148 601–621.
• Skorokhod, A. V. (1956). Limit theorems for stochastic processes., Theory Probab. Appl. 1 261.
• van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer-Verlag, New York. With applications to statistics.
• Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood., J. Amer. Statist. Assoc. 97 508–513.
• Walther, G. (2009). Inference and modeling with log-concave distributions., Statist. Sci. 24 319–327.
• Watanabe, H. (1970). An asymptotic property of Gaussian processes., Trans. Amer. Math. Soc. 148 233.
• Whitt, W. (1970). Weak convergence of probability measures on the function space $C[0,\infty )$., Ann. Math. Stat. 41 939–944.
• Whitt, W. (1980). Some useful functions for functional limit theorems., Math. Oper. Res. 5 67–85.
• Whitt, W. (2002)., Stochastic-Process Limits. Springer Series in Operations Research. Springer-Verlag, New York.
• Xu, M. and Samworth, R. J. (2017). High-dimensional nonparametric density estimation via symmetry and shape constraints. Technical Report, Available at, http://www.statslab.cam.ac.uk/~rjs57/Research.html.