On Boundary Correction in Kernel Estimation of ROC Curves
DOI:
https://doi.org/10.17713/ajs.v38i1.257Abstract
The Receiver Operating Characteristic (ROC) curve is a statistical tool for evaluating the accuracy of diagnostics tests. The empirical ROC curve (which is a step function) is the most commonly used non-parametric estimator for the ROC curve. On the other hand, kernel smoothing methods have been used to obtain smooth ROC curves. The preceding process is based on kernel estimates of the distribution functions. It has been observedthat kernel distribution estimators are not consistent when estimating a distribution function near the boundary of its support. This problem is due to “boundary effects” that occur in nonparametric functional estimation. To avoid these difficulties, we propose a generalized reflection method of boundary correction in the estimation problem of ROC curves. The proposed method generates a class of boundary corrected estimators.
References
Azzalini, A. (1981). A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika, 68, 326-328.
Horová, I., Koláček, J., Zelinka, J., and El-Shaarawi, A. H. (2008). Smooth estimates of distribution functions with application in environmental studies. Advanced topics on mathematical biology and ecology, 122-127.
Karunamuni, R. J., and Alberts, T. (2005a). A generalized reflection method of boundary correction in kernel density estimation. Canadian Journal of Statistics, 33, 497-509.
Karunamuni, R. J., and Alberts, T. (2005b). On boundary correction in kernel density estimation. Statistical Methodology, 2, 191-212.
Karunamuni, R. J., and Alberts, T. (2006). A locally adaptive transformation method of boundary correction in kernel density estimation. Journal of Statistical Planning and Inference, 136, 2936-2960.
Karunamuni, R. J., and Zhang, S. (2008). Some improvements on a boundary corrected kernel density estimator. Statistics & Probability Letters, 78, 497-507.
Lejeune, M., and Sarda, P. (1992). Smooth estimators of distribution and density functions. Computational Statistics & Data Analysis, 14, 457-471.
Lloyd, C. J. (1998). The use of smoothed ROC curves to summarise and compare diagnostic systems. Journal of the American Statistical Association, 93, 1356-1364.
Lloyd, C. J., and Yong, Z. (1999). Kernel estimators of the ROC curve are better than empirical. Statistics and Probability Letters, 44, 221-228.
Nadaraya, E. A. (1964). Some new estimates for distribution functions. Theory of Probability and its Application, 15, 497-500.
Reiss, R. D. (1981). Nonparametric estimation of smooth distribution functions. Scandinavian Journal of Statistics, 8, 116-119.
Sheather, S. J., and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B, 53, 683-690.
Silverman, W. R. (1986). Density Estimation for Statistics and Data Analysis. London: Chapman and Hall.
Wand, M. P., and Jones, M. C. (1995). Kernel Smoothing. London: Chapman and Hall.
Zhang, S., and Karunamuni, R. J. (1998). On kernel density estimation near endpoints. J. Statist. Planning and Inference, 70, 301–316.
Zhang, S., and Karunamuni, R. J. (2000). On nonparametric density estimation at the boundary. Nonparametric Statistics, 12, 197–221.
Zhang, S., Karunamuni, R. J., and Jones, M. C. (1999). An improved estimator of the density function at the boundary. Journal of the American Statistical Association, 94, 1231–1241.
Downloads
Published
How to Cite
Issue
Section
License
The Austrian Journal of Statistics publish open access articles under the terms of the Creative Commons Attribution (CC BY) License.
The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed.
Copyright on any research article published by the Austrian Journal of Statistics is retained by the author(s). Authors grant the Austrian Journal of Statistics a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified.
Manuscripts should be unpublished and not be under consideration for publication elsewhere. By submitting an article, the author(s) certify that the article is their original work, that they have the right to submit the article for publication, and that they can grant the above license.