title: Tolerance limits for a ratio of normal random variables.
abstract: 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.
authors.affiliation: Department of Biostatistics, MedImmune, Inc., Gaithersburg, Maryland 20878, USA. zhangla@MedImmune.com
authors.forename: Lanju
authors.initials: L
authors.lastname: Zhang
authors.forename: Thomas
authors.initials: T
authors.lastname: Mathew
authors.forename: Harry
authors.initials: H
authors.lastname: Yang
authors.forename: K
authors.initials: K
authors.lastname: Krishnamoorthy
authors.forename: Iksung
authors.initials: I
authors.lastname: Cho
mesh_terms: D016001:Confidence Intervals; D015233:Models, Statistical; D016011:Normal Distribution; D011897:Random Allocation