Estimating Mean Concentrations under Transformation for Environmental Data with Detection Limits

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Abstract
The reporting procedures for potentially toxic pollutants are complicated by the fact that concentrations are measured using small samples that include a number of observations lying below some detection limit. Furthermore, there is often a small number of high concentrations observed in combination with a substantial number of low concentrations. This results in small, nonnormally distributed censored samples. This article presents maximum likelihood estimators for the mean of a population, based on censored samples that can be transformed to normality. The method estimates the optimal power transformation in the Box-Cox family by searching the censored-data likelihood. Maximum likelihood estimators for the mean in the transformed scale are calculated via the expectation-maximization algorithm. Estimates for the mean in the original scale are functions of the estimated mean and variance in the transformed population. Confidence intervals are computed using the delta method and the nonparametric percentile and bias-corrected percentile versions of Efron's bootstrap. A simulation study over sampling configurations expected with environmental data indicates that the delta method, combined with a reliable value for the power transformation, produces intervals with better coverage properties than the bootstrap intervals.

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