Confidence intervals for probabilities of exceeding threshold limits with censored log-normal data

In enforcing safety regulations in the context of occupational health, the question of the probability of exceeding the so-called threshold limit value (TLV) in homogeneous exposure groups (HEG) arises. Because data are usually few, a simple non-parametric approach based on numbers of exceedances over the TLV cannot be usefully adopted. However, the distribution of airborne concentrations in the exposures within HEGs is often compatible with a log-normal distribution. Therefore inference based on this distribution may be adopted. There remain two problems for which there has been no satisfactory solution: the treatment of left censoring due to the detection limits of the exposure determinations, and the absence of a confidence interval (CI) for the probability of exceedance. In the non-censored case we present an exact. confidence interval based on the non-central T-distribution and two Monte Carlo solutions based on the bootstrap and on the Gibbs sampler. The Monte Carlo methods are extended in various ways to treat the censored case. A large scale simulation study shows that the Gibbs Sampling CIs has coverage close to nominal in nearly all situations. The parametric bootstrap, after bias correction, performs nearly as well. We also present some extensions of the Gibbs sampling method to the analysis of fibre exposure data, and to the analysis of the effect of finite precision and measurement error. A worked example is presented.