Confidence Estimation of the Cross-Product Ratio of Binomial Proportions under Different Sampling Schemes

We consider a general problem of the interval estimation for a cross-product ratio $$\rho=\frac{p_{1}(1-p_{2})}{p_{2}(1-p_{1})}$$ according to data from two independent samples. Each sample may be obtained in the framework of direct or inverse Binomial sampling schemes. Asymptotic confidence intervals are constructed in accordance with different types of sampling schemes, with parameter estimators demonstrating exponentially decreasing bias. Our goal is to investigate the cases when the normal approximations (which are relatively simple) for estimators of the cross-product ratio are reliable for the construction of confidence intervals. We use the closeness of the confidence coefficient to the nominal confidence level as our main evaluation criterion, and use the Monte-Carlo method to investigate the key probability characteristics of intervals corresponding to all possible combinations of sampling schemes. We present estimations of the coverage probability, expectation and standard deviation of interval widths in tables and provide some recommendations for applying each obtained interval.