Estimating Receiver Operating Characteristic Curve (ROC) Using Birnbaum-Saunders Kernel

Document Type : Original Paper


Department of Statistics, Faculty of Mathematical Sciences & Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran


Many researchers use the receiver operating characteristic curve (ROC) as a popular way of displaying, evaluating and comparing the discriminatory accuracy of diagnostic tests. The most common

approach for estimating the ROC curve is using nonparametric kernel estimates in two parts, sensitivity and specificity. Kernel estimators, however, at the beginning and end points of the data domain, known as boundary points, have a slower convergence rate than other points in the domain and are not convergent to the actual value of the probability distribution. This problem is known as the boundary problem. One way to solve the boundary problem in kernel estimators is to use asymmetric kernels. This paper proposes a new kernel estimator for the ROC curve based on the asymmetric Birnbaum-Saunders (B-S) kernel and the asymptotic convergence of the proposed estimator is shown. In addition, the analytical superiority of the proposed estimator over the corresponding symmetric kernel-type estimator is shown. The performance

of the proposed estimator is illustrated via a numerical study. The results show that the proposed estimator outperforms the other commonly-used methods. The application of the proposed method to a set of medical data is also presented.


Main Subjects

[1] Altman, N, Leger, C (1995) Bandwidth selection for kernel distribution function estimation. J Stat PlanInference, 46, 195-214.
[2] Chen, SX (1999) Beta kernel estimators for density functions. Comput Stat Data Anal, 31, 131-145.[3] Chen, SX (2000) Probability density function estimation using gamma kernels. Ann Inst Stat Math,52, 471-480.
[4] Du P, Tang L (2009) Transformation-invariant and nonparametric monotone smooth estimation ofROC curves. Stat Med, 28, 349-359.
[5] Duong, T. (2016). Non-parametric smoothed estimation of multivariate cumulative distribution an survival functions, and receiver operating characteristic curves. J Korean Stat Soc, 45 (1), 33-50.
[6] Fawcett T (2006) An introduction to ROC analysis. Pattern Recognition Lett, 27, 861-874. Green DM,Swets JA (1966) Signal detection theory and psychophysics. Wiley New York.
[7] Green DM, Swets JA (1966) Signal detection theory and psychophysics. Wiley New York.
[8] Hans P, Albert A, Born J, Chapelle JP (1985) Derivation of a bioclinical prognostic index in severehead injury. Intensive Care Med, 11, 186-191.
[9] Horová I, Koláček J, Zelinka J, El-Shaarawi AH (2008) Smooth estimates of distribution functionswith application in environmental studies. Article in Proceedings. Advanced topics on mathematicalbiology and ecology, 1, 122-127.
[10] Hsieh F, Turnbull BW (1996) Nonparametric and semiparametric estimation of the receiver operatingcharacteristic curve. Ann. Stat., 24, 25-40.
[11] Koláček J, Karunamuni RJ (2009) On boundary correction in kernel estimation of ROC curves. AustrianJ. Stat., 38, 17–32-17–32.
[12] Lafaye de Micheaux, P., & Ouimet, F. (2021). A study of seven asymmetric kernels for the estimationof cumulative distribution functions. Mathematics, 9 (20), 2605.
[13] Lasko TA. Bhagwat JG, Zou KH, Ohno-Machado L (2005) The use of receiver operating characteristiccurves in biomedical informatics. J. Biomed. Inform., 38, 404-415.
[14] Lloyd CJ (1998) Using smoothed receiver operating characteristic curves to summarize and comparediagnostic systems. J Am Stat Assoc, 93, 1356-1364.
[15] Lloyd CJ, Yong Z (1999) Kernel estimators of the ROC curve are better than empirical. Stat ProbabLett, 44, 221-228.
[16] Marchant C, Bertin K, Leiva V, Saulo H (2013) Generalized Birnbaum–Saunders kernel density
estimators and an analysis of financial data. Comput Stat Data Anal, 63, 1-15.
[17] Mansouri, B., Atiyah Sayyid Al-Farttosi, S., Mombeni, H., & Chinipardaz, R. (2022). Estimating
Cumulative Distribution Function Using Gamma Kernel. J. Sci. Islam. Repub. 33 (1), 45-54.
[18] Mombeni HA, Mansouri B, Akhoond MR (2021) Asymmetric kernels for boundary modification indis-tribution function estimation. Revstat Stat. J.
[19] Pulit M (2016) A new method of kernel-smoothing estimation of the ROC curve. Metrika, 79, 603-634.
[20] Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall: London.[21] Tang L, Du P, Wu C (2010) Compare diagnostic tests using transformation-invariant smoothed ROCcurves. J Stat Plan Inference, 140, 3540-3551.
[22] Tenreiro C (2013) Boundary kernels for distribution function estimation. Revstat Stat. J., 11, 169-190.
[23] Tenreiro C (2018) A new class of boundary kernels for distribution function estimation. Commun.Stat. Theory Methods, 47, 5319-5332.
[24] Wasserman L (2006) All of Nonparametric Statistics, Springer: New York.
[25] Zhang S, Karunamuni RJ, Jones MC (1999) An improved estimator of the density function at theboundary. J Am Stat Assoc, 94, 1231-1240.
[26] Zhou XH,. McClish DK, Obuchowski NA (2009) Statistical Methods in Diagnostic Medicine. JohnWiley & Sons.
[27] Zou KH. Hall W, Shapiro DE (1997) Smooth non-parametric receiver operating characteristic (ROC)curves for continuous diagnostic tests. Stat Med, 16, 2143-2156.