An integrated-likelihood-ratio confidence interval for a proportion based on underreported and infallible data

We derive and examine the interval width and coverage properties of an integrated-likelihood-ratio confidence interval for the binomial parameter p using a double-sampling scheme. The data consist of a relatively large fallible sample containing underreported data and a relatively small infallible subsample. Via Monte Carlo simulations, we determine that the new integrated-likelihood-ratio interval estimator displays slightly conservative to moderately conservative coverage properties for small to medium sample sizes and can have shorter average-interval width than two previously proposed confidence intervals when p < 0.10 or p > 0.90. We also apply the integrated-likelihood-ratio confidence interval to a real-data set and determine that the integrated-likelihood-ratio interval has superior performance when contrasted to two properties of two competing confidence intervals.