In diagnostic studies, both sensitivity and specificity depend on cut-off point and they are well-known measures for diagnostic accuracy. The diagnostic cut-off point is mostly unknown and needs to be determined by some optimization criteria out of which the one based on the Youden index has been widely adopted in practice. The estimation of the optimal cut-off point associated with Youden index depends on both diseased and healthy samples, henceforth, sensitivity and specificity at the estimated cut-off point are correlated. Therefore, it is desirable to make joint inference on both sensitivity and specificity at the estimated cut-off point. Several parametric and non-parametric approaches are proposed to estimate the joint confidence region of sensitivity and specificity at the cut-off point determined by the Youden index. A real data set is analyzed using the proposed approaches.