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Kernel estimators of the ROC curve are better than empirical

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  • Lloyd, Chris J.
  • Yong, Zhou

Abstract

The receiver operating characteristic (ROC) is a curve used to summarise the performance of a binary decision rule. It can be expressed in terms of the underlying distributions functions of the diagnostic measurement that underlies the rule. Lloyd (1998) has proposed estimating the ROC curve from kernel smoothing of these distribution functions and has presented asymptotic formulas for the bias and standard deviation of the resulting curve estimator. This paper compares the asymptotic accuracy of the kernel-based estimator with the fully empirical estimator. It is shown that the empirical estimator is deficient compared to the kernel estimator and that this deficiency is unbounded as sample size increases. A simulation study using both unimodal and bimodal distributions indicates that the gains in accuracy are significant for realistic sample sizes. Kernel-based ROC estimators can now be recommended.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:44:y:1999:i:3:p:221-228
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    1. M. Falk, 1983. "Relative efficiency and deficiency of kernel type estimators of smooth distribution functions," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 37(2), pages 73-83, June.
    2. Donald Dorfman & Edward Alf, 1968. "Maximum likelihood estimation of parameters of signal detection theory—A direct solution," Psychometrika, Springer;The Psychometric Society, vol. 33(1), pages 117-124, March.
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    Cited by:

    1. Gong, Yun & Peng, Liang & Qi, Yongcheng, 2010. "Smoothed jackknife empirical likelihood method for ROC curve," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1520-1531, July.
    2. Alicja Jokiel-Rokita & Rafał Topolnicki, 2019. "Minimum distance estimation of the binormal ROC curve," Statistical Papers, Springer, vol. 60(6), pages 2161-2183, December.
    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. Douglas Mossman & Hongying Peng, 2016. "Using Dual Beta Distributions to Create “Proper†ROC Curves Based on Rating Category Data," Medical Decision Making, , vol. 36(3), pages 349-365, April.
    5. Yang, Hanfang & Zhao, Yichuan, 2013. "Smoothed jackknife empirical likelihood inference for the difference of ROC curves," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 270-284.
    6. Kang, Le & Tian, Lili, 2013. "Estimation of the volume under the ROC surface with three ordinal diagnostic categories," Computational Statistics & Data Analysis, Elsevier, vol. 62(C), pages 39-51.
    7. Michał Pulit, 2016. "A new method of kernel-smoothing estimation of the ROC curve," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(5), pages 603-634, July.
    8. Yang, Hanfang & Zhao, Yichuan, 2015. "Smoothed jackknife empirical likelihood inference for ROC curves with missing data," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 123-138.
    9. Chen, Xiwei & Vexler, Albert & Markatou, Marianthi, 2015. "Empirical likelihood ratio confidence interval estimation of best linear combinations of biomarkers," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 186-198.
    10. Yang, Hanfang & Zhao, Yichuan, 2012. "Smoothed empirical likelihood for ROC curves with censored data," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 254-263.
    11. Gaëlle Chagny & Claire Lacour, 2015. "Optimal adaptive estimation of the relative density," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 605-631, September.
    12. Lloyd, Chris J., 2002. "Estimation of a convex ROC curve," Statistics & Probability Letters, Elsevier, vol. 59(1), pages 99-111, August.
    13. Arís Fanjul-Hevia & Wenceslao González-Manteiga, 2018. "A comparative study of methods for testing the equality of two or more ROC curves," Computational Statistics, Springer, vol. 33(1), pages 357-377, March.
    14. Funke, Benedikt & Palmes, Christian, 2017. "A note on estimating cumulative distribution functions by the use of convolution power kernels," Statistics & Probability Letters, Elsevier, vol. 121(C), pages 90-98.
    15. Yousef, Waleed A. & Kundu, Subrata & Wagner, Robert F., 2009. "Nonparametric estimation of the threshold at an operating point on the ROC curve," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 4370-4383, October.
    16. Elisa–María Molanes-López & Ricardo Cao, 2008. "Relative density estimation for left truncated and right censored data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 693-720.
    17. 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.

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