IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v75y2014icp248-264.html
   My bibliography  Save this article

Computing confidence intervals for log-concave densities

Author

Listed:
  • Azadbakhsh, Mahdis
  • Jankowski, Hanna
  • Gao, Xin

Abstract

In Balabdaoui, Rufibach, and Wellner (2009), pointwise asymptotic theory was developed for the nonparametric maximum likelihood estimator of a log-concave density. Here, the practical aspects of their results are explored. Namely, the theory is used to develop pointwise confidence intervals for the true log-concave density. To do this, the quantiles of the limiting process are estimated and various ways of estimating the nuisance parameter appearing in the limit are studied. The finite sample size behavior of these estimated confidence intervals is then studied via a simulation study of the empirical coverage probabilities.

Suggested Citation

  • Azadbakhsh, Mahdis & Jankowski, Hanna & Gao, Xin, 2014. "Computing confidence intervals for log-concave densities," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 248-264.
    • Handle: RePEc:eee:csdana:v:75:y:2014:i:c:p:248-264
    DOI: 10.1016/j.csda.2014.01.020
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947314000334
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2014.01.020?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Moulinath Banerjee & Jon A. Wellner, 2005. "Confidence Intervals for Current Status Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 405-424, September.
    2. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.
    3. Piet Groeneboom & Geurt Jongbloed & Jon A. Wellner, 2008. "The Support Reduction Algorithm for Computing Non‐Parametric Function Estimates in Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 385-399, September.
    4. Walther G., 2002. "Detecting the Presence of Mixing with Multiscale Maximum Likelihood," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 508-513, June.
    5. Dümbgen, Lutz & Rufibach, Kaspar, 2011. "logcondens: Computations Related to Univariate Log-Concave Density Estimation," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 39(i06).
    6. Chang, George T. & Walther, Guenther, 2007. "Clustering with mixtures of log-concave distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6242-6251, August.
    7. Duong, Tarn, 2007. "ks: Kernel Density Estimation and Kernel Discriminant Analysis for Multivariate Data in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 21(i07).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dümbgen, Lutz & Wellner, Jon A. & Wolff, Malcolm, 2016. "A law of the iterated logarithm for Grenander’s estimator," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3854-3864.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dümbgen, Lutz & Mösching, Alexandre & Strähl, Christof, 2021. "Active set algorithms for estimating shape-constrained density ratios," Computational Statistics & Data Analysis, Elsevier, vol. 163(C).
    2. Durot, Cécile & Huet, Sylvie & Koladjo, François & Robin, Stéphane, 2013. "Least-squares estimation of a convex discrete distribution," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 282-298.
    3. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.
    4. Feng, Oliver Y. & Chen, Yining & Han, Qiyang & Carroll, Raymond J & Samworth, Richard J., 2022. "Nonparametric, tuning-free estimation of S-shaped functions," LSE Research Online Documents on Economics 111889, London School of Economics and Political Science, LSE Library.
    5. Balabdaoui, Fadoua & Kulagina, Yulia, 2020. "Completely monotone distributions: Mixing, approximation and estimation of number of species," Computational Statistics & Data Analysis, Elsevier, vol. 150(C).
    6. Hazelton, Martin L., 2011. "Assessing log-concavity of multivariate densities," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 121-125, January.
    7. Hu, Hao & Yao, Weixin & Wu, Yichao, 2017. "The robust EM-type algorithms for log-concave mixtures of regression models," Computational Statistics & Data Analysis, Elsevier, vol. 111(C), pages 14-26.
    8. Hu, Hao & Wu, Yichao & Yao, Weixin, 2016. "Maximum likelihood estimation of the mixture of log-concave densities," Computational Statistics & Data Analysis, Elsevier, vol. 101(C), pages 137-147.
    9. Yining Chen & Richard J. Samworth, 2016. "Generalized additive and index models with shape constraints," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 729-754, September.
    10. Ge, Suqin & Macieira, João, 2020. "Unobserved Worker Quality and Inter-Industry Wage Differentials," GLO Discussion Paper Series 491, Global Labor Organization (GLO).
    11. Mu, Xiaosheng, 2015. "Log-concavity of a mixture of beta distributions," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 125-130.
    12. Battey, Heather & Linton, Oliver, 2014. "Nonparametric estimation of multivariate elliptic densities via finite mixture sieves," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 43-67.
    13. repec:jss:jstsof:36:i02 is not listed on IDEAS
    14. Yining Chen, 2015. "Semiparametric Time Series Models with Log-concave Innovations: Maximum Likelihood Estimation and its Consistency," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(1), pages 1-31, March.
    15. R. C. Rodríguez-Caro & E. Graciá & S. P. Blomberg & H. Cayuela & M. Grace & C. P. Carmona & H. A. Pérez-Mendoza & A. Giménez & R. Salguero-Gómez, 2023. "Anthropogenic impacts on threatened species erode functional diversity in chelonians and crocodilians," Nature Communications, Nature, vol. 14(1), pages 1-10, December.
    16. Elina Robeva & Bernd Sturmfels & Ngoc Tran & Caroline Uhler, 2021. "Maximum likelihood estimation for totally positive log‐concave densities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(3), pages 817-844, September.
    17. Pramita Bagchi & Subhra Sankar Dhar, 2020. "A study on the least squares estimator of multivariate isotonic regression function," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(4), pages 1192-1221, December.
    18. Chang, George T. & Walther, Guenther, 2007. "Clustering with mixtures of log-concave distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6242-6251, August.
    19. Guillermo Basulto-Elias & Alicia L. Carriquiry & Kris Brabanter & Daniel J. Nordman, 2021. "Bivariate Kernel Deconvolution with Panel Data," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 122-151, May.
    20. Rufibach Kaspar, 2012. "A Smooth ROC Curve Estimator Based on Log-Concave Density Estimates," The International Journal of Biostatistics, De Gruyter, vol. 8(1), pages 1-29, April.
    21. Mei-Cheng Wang & Yuxin Zhu, 2022. "Bias correction via outcome reassignment for cross-sectional data with binary disease outcome," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 28(4), pages 659-674, October.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:75:y:2014:i:c:p:248-264. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.