The Youden index serves as an excellent summary statistic of a Receiver Operating Characteristic (ROC) curve as it directly reflects the misclassification rate. The inference of the Youden index usually involves the grid search of the optimal cutoff and could be even complicated if the biomarker measurement is subject to a limit of detection. In this talk, the derivation of a smoothed empirical likelihood (SEL) will be illustrated, for the inference of the Youden index in the presence of incomplete data caused by a limit of detection. The method allows nonparametric profile likelihood ratio test and avoids the optimal cutoff estimation. Both asymptotic properties and empirical performance of the SEL method will be evaluated. Furthermore, the feasibility of using the proposed method as a Kolmogorov-Smirnov two-sample test procedure under random censorship will also be discussed.