Logarithmic confidence intervals for the cross-product ratio of binomial proportions under different sampling schemes

Abstract

We consider the problem of logarithmic interval estimation for a cross-product ratio $\rho =\frac{p_1(1-p_2)}{p_2(1-p_1)}$ with data from two independent Bernoulli samples. Each sample may be obtained in the framework of direct or inverse Binomial sampling schemes. Asymptotic logarithmic confidence intervals are constructed under different types of sampling schemes, with parameter estimators demonstrating exponentially decreasing bias. Our goal is to investigate the cases when the relatively simple normal approximations for estimators of the cross-product ratio are reliable for constructing logarithmic 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. Also, we provide some recommendations for applying each logarithmic interval obtained.

Keywords:

• Cross-product ratio
• Direct binomial sampling scheme
• Inverse binomial sampling scheme
• Logarithmic confidence interval
• Normal approximation

Acknowledgements

The authors are grateful to the reviewers for carefully reading the manuscript and for offering substantial comments and suggestions which enabled them to improve the presentation. The last listed author’s research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project no. 1.13556.2019/13.1.