The Area Under the Receiving Operating Characteristic Curve (AUC) is frequently used for assessing the overall accuracy of a diagnostic marker. However, estimation of AUC relies on knowledge of the true outcomes of subjects: diseased or non-diseased. Because disease verification based on a gold standard is often expensive and/or invasive, only a limited number of patients are sent to verification at doctors' discretion. Estimation of AUC is generally biased if only small verified samples are used and it is thus necessary to make corrections for such lack of information. However, correction based on the ignorable missingness assumption (or missing at random) is also biased if the missing mechanism indeed depends on the unknown disease outcome, which is called nonignorable missing. In this paper, we propose a propensity-score-adjustment method for estimating AUC based on the instrumental variable assumption when the missingness of disease status is nonignorable. The new method makes parametric assumptions on the verification probability, and the probability of being diseased for verified samples rather than for the whole sample. The proposed parametric assumption on the observed sample is easier to be verified than the parametric assumption on the full sample. We establish the asymptotic properties of the proposed estimators. A simulation study is performed to compare the proposed method with existing methods. The proposed method is also applied to an Alzheimer's disease data collected by National Alzheimer's Coordinating Center.