Electronic Journal of Statistics

Maximum-likelihood estimation of a log-concave density based on censored data

Lutz Dümbgen, Kaspar Rufibach, and Dominic Schuhmacher

Full-text: Open access

Abstract

We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at $+\infty$, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples.

Article information

Source
Electron. J. Statist. Volume 8, Number 1 (2014), 1405-1437.

Dates
First available in Project Euclid: 20 August 2014

Permanent link to this document
http://projecteuclid.org/euclid.ejs/1408540292

Digital Object Identifier
doi:10.1214/14-EJS930

Mathematical Reviews number (MathSciNet)
MR3263127

Zentralblatt MATH identifier
1298.62062

Subjects
Primary: 62G07: Density estimation 62N01: Censored data models 62N02: Estimation 65C60: Computational problems in statistics

Keywords
Active set algorithm binning cure parameter expectation-maximization algorithm interval-censoring qualitative constraints right-censoring

Citation

Dümbgen, Lutz; Rufibach, Kaspar; Schuhmacher, Dominic. Maximum-likelihood estimation of a log-concave density based on censored data. Electron. J. Statist. 8 (2014), no. 1, 1405--1437. doi:10.1214/14-EJS930. http://projecteuclid.org/euclid.ejs/1408540292.


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References

  • [1] Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum-likelihood from incomplete data via the EM algorithm (with discussion). J. Royal Statist. Soc. Ser. B 39(1) 1–38.
  • [2] Dümbgen, L., Freitag, S. and Jongbloed, G. (2004). Consistency of concave regression with an application to current-status data. Math. Meth. Statist. 13 69–81.
  • [3] Dümbgen, L., Freitag-Wolf, S. and Jongbloed, G. (2006). Estimating a unimodal distribution from interval-censored data. J. Amer. Statist. Assoc. 101 1094–1106.
  • [4] Dümbgen, L., Hüsler, A. and Rufibach, K. (2007, revised 2011a). Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, University of Bern.
  • [5] 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(1) 40–68.
  • [6] Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39(2) 702–730.
  • [7] Edmunson, J. H., Fleming, T. R., Decker, D. G., Malkasian, G. D., Jefferies, J. A., Webb, M. J. and Kvols, L. K. (1979). Different chemotherapeutic sensitivities and host factors affecting prognosis in advanced ovarian carcinoma vs. minimal residual disease. Cancer Treatment Reports 63 241–247.
  • [8] Fay, M. P. (2013). Interval: Weighted Logrank Tests and NPMLE for Interval Censored Data. R package, available at http://cran.r-project.org/web/packages/interval/.
  • [9] R Core Team (2013). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Available at http://www.r-project.org/.
  • [10] Schuhmacher, D., Hüsler, A. and Dümbgen, L. (2011). Multivariate log-concave distributions as a nearly parametric model. Statist. Risk Modeling 28(3) 277–295.
  • [11] Schuhmacher, D., Rufibach, K. and Dümbgen, L. (2013). Logconcens: Maximum Likelihood Estimation of a Log-Concave Density Based on Censored Data. R package, available at http://cran.r-project.org/web/packages/logconcens/.
  • [12] Silverman, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method. Ann. Statist. 10(3) 795–810.
  • [13] Therneau, T. (2013). Survival: Survival Analysis. R package, available at http://cran.r-project.org/web/packages/survival/.
  • [14] Turnbull, B. W. (1976). The empirical distribution function with arbitrarily grouped, censored and truncated data. J. Roy. Statist. Soc. Ser. B 38(3) 290–295.
  • [15] Walther, G. (2009). Inference and modeling with log-concave distributions. Statist. Sci. 24(3) 319–327.