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A Smooth ROC Curve Estimator Based on Log-Concave Density Estimates

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  • Rufibach Kaspar

    (University of Zurich)

Abstract

We introduce a new smooth estimator of the ROC curve based on log-concave density estimates of the constituent distributions. We show that our estimate is asymptotically equivalent to the empirical ROC curve if the underlying densities are in fact log-concave. In addition, we empirically show that our proposed estimator exhibits an efficiency gain for finite sample sizes with respect to the standard empirical estimate in various scenarios and that it is only slightly less efficient, if at all, compared to the fully parametric binormal estimate in case the underlying distributions are normal. The estimator is also quite robust against modest deviations from the log-concavity assumption. We show that bootstrap confidence intervals for the value of the ROC curve at a fixed false positive fraction based on the new estimate are on average shorter compared to the approach by Zhou and Qin (2005), while maintaining coverage probability. Computation of our proposed estimate uses the R package logcondens that implements univariate log-concave density estimation and can be done very efficiently using only one line of code. These obtained results lead us to advocate our estimate for a wide range of scenarios.

Suggested Citation

  • 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.
    • Handle: RePEc:bpj:ijbist:v:8:y:2012:i:1:n:7
    DOI: 10.1515/1557-4679.1378
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    References listed on IDEAS

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    1. Peng, Roger, 2008. "Caching and Distributing Statistical Analyses in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 26(i07).
    2. Cule, Madeleine & Gramacy, Robert B. & Samworth, Richard, 2009. "LogConcDEAD: An R Package for Maximum Likelihood Estimation of a Multivariate Log-Concave Density," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 29(i02).
    3. Hall, Peter G. & Hyndman, Rob J., 2003. "Improved methods for bandwidth selection when estimating ROC curves," Statistics & Probability Letters, Elsevier, vol. 64(2), pages 181-189, August.
    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. Lloyd, Chris J. & Yong, Zhou, 1999. "Kernel estimators of the ROC curve are better than empirical," Statistics & Probability Letters, Elsevier, vol. 44(3), pages 221-228, September.
    6. Lloyd, Chris J., 2002. "Estimation of a convex ROC curve," Statistics & Probability Letters, Elsevier, vol. 59(1), pages 99-111, August.
    7. Liansheng Tang & Scott S. Emerson & Xiao-Hua Zhou, 2008. "Nonparametric and Semiparametric Group Sequential Methods for Comparing Accuracy of Diagnostic Tests," Biometrics, The International Biometric Society, vol. 64(4), pages 1137-1145, December.
    8. Hazelton, Martin L., 2011. "Assessing log-concavity of multivariate densities," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 121-125, January.
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    Cited by:

    1. 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.

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