Tolerance limits and tolerance intervals for ratios of normal random variables using a bootstrap calibration

Area of Science:

• Statistics
• Biostatistics
• Health Economics

Background:

• Deriving tolerance limits and intervals for ratios of random variables is crucial in various scientific fields.
• Existing methods may lack accuracy for specific distributions like bivariate normal or lognormal/normal.

Purpose of the Study:

• To develop and validate a novel methodology for one-sided tolerance limits and two-sided tolerance intervals for ratios of random variables.
• To extend the methodology for computing confidence limits for the median of the ratio random variable.
• To demonstrate the accuracy and applicability of the proposed approach through numerical results and real-world examples.

Main Methods:

• The study employs a nonparametric tolerance limit approach based on a parametric bootstrap sample.
• Bootstrap calibration is utilized to enhance the accuracy of the derived limits and intervals.
• The methodology is adapted for calculating confidence limits for the median of the ratio random variable.

Main Results:

• Numerical results confirm the accuracy of the proposed bootstrap-based methodology.
• The approach effectively derives one-sided tolerance limits and two-sided tolerance intervals for ratio distributions.
• Confidence limits for the median of the ratio are accurately computed.

Conclusions:

• The developed methodology provides an accurate and reliable approach for tolerance limit and interval estimation for ratio random variables.
• This method is applicable to distributions such as bivariate normal and lognormal/normal.
• The study highlights the utility of the methodology in radioactivity count analysis and health economics cost-effectiveness analysis.