In medical diagnostics, the ROC curve is the graph of sensitivity against 1-specificity as the diagnostic threshold runs through all possible values. The ROC curve and its associated summary indices are very useful for the evaluation of the discriminatory ability of biomarkers/diagnostic tests with continuous measurements. Among all summary indices, the area under the ROC curve (AUC) is the most popular diagnostic accuracy index, which has been extensively used by researchers for biomarker evaluation and selection. Sometimes, taking the actual measurements of a biomarker is difficult and expensive, whereas ranking them without actual measurements can be easy. In such cases, ranked set sampling based on judgment order statistics would provide more representative samples yielding more accurate estimation. In this study, Gaussian kernel is utilized to obtain a nonparametric estimate of the AUC. Asymptotic properties of the AUC estimates are derived based on the theory of U-statistics. Intensive simulation is conducted to compare the estimates using ranked set samples versus simple random samples. The simulation and theoretical derivation indicate that ranked set sampling is generally preferred with smaller variances and mean squared errors (MSE). The proposed method is illustrated via a real data analysis.