Tolerance limits for a ratio of normal random variables

The problem of deriving an upper tolerance limit for a ratio of two normally distributed random variables is addressed, when the random variables follow a bivariate normal distribution, or when they are independent normal. The derivation uses the fact that an upper tolerance limit for a random variable can be derived from a lower confidence limit for the cumulative distribution function (cdf) of the random variable. The concept of a generalized confidence interval is used to derive the required lower confidence limit for the cdf. In the bivariate normal case, a suitable representation of the cdf of the ratio of the marginal normal random variables is also used, coupled with the generalized confidence interval idea. In addition, a simplified derivation is presented in the situation when one of the random variables has a small coefficient of variation. The problem is motivated by an application from a reverse transcriptase assay. Such an example is used to illustrate our results. Numerical results are also reported regarding the performance of the proposed tolerance limit.