With the progressive technology, many medical researches are aimed to develop diagnostic tests that can detect diseases faster and accurately. The assessment of the accuracy of the diagnostic test for classifying two groups is through the receiver operating characteristic (ROC) curve and the area under the ROC curve (AUC). When a paired design is considered, the sample size determination requires the information about two AUC estimates and the corresponding variance and covariance of two AUC estimators. This paper derives the nonparametric estimators of the variance and covariance of two AUC estimators. The result is used to derive the sample size formula when the paired sample is planned. Since most of the results do not have a closed form, numerical results are provided under various scenarios.