Exact interval estimation for three parameters subject to false positive misclassification

SummaryBinary data subject to one type of misclassification exist in various fields. It is collected in a double‐sampling scheme that includes a gold standard test and a fallible test. The main parameter of interest for this type of data is the positive probability of the gold standard test. Existing intervals are unreliable because the given nominal level is not achieved. In this paper, we construct an exact interval by inverting the E+M score tests and improve it by the general ‐function method. We find that the total length of the improved interval is shorter than the exact intervals that are also the improved intervals when we apply the ‐function to several existing approximate intervals, including the score and Bayesian intervals. Therefore, it is recommended for practice. We are also interested in two other parameters: —the difference between the two positive rates for the fallible and gold standard tests—and —the false positive rate for the fallible test. To the best of our knowledge, the research on these two parameters is limited. For , we find that any interval for can be converted to an interval for . So, the interval converted from the aforementioned recommended interval for is recommended for inferring . For , the improved interval by the ‐function method over the E+M score interval is derived. We use an example to illustrate how the intervals are computed and provide a real data analysis.