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Fitting human exposure data with the Johnson SB distribution

Journal of Exposure Science & Environmental Epidemiology volume 16, pages 56–62 (2006)Cite this article

A Corrigendum to this article was published on 15 July 2006

Abstract

Exposure evaluations for epidemiological investigations and risk assessments may require estimates of background concentrations and peak exposures, as well as the population mean and variance. The SB distribution is a theoretically appealing probability function for characterizing ratios, and random variables bound by extremes, such as human exposures and environmental concentrations. However, fitting the parameters of this distribution with maximum likelihood methods is often problematic, and some alternative methods are examined here. Two methods based on percentiles, a quantile estimator, and a method-of-moments fitting procedure are explored. The quantile and method-of-moments procedures are based on new explicit expressions for the first four moments of this distribution. The fitting procedures are compared by simulation, and with actual data sets consisting of measurements of human exposure to airborne contaminants.

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Acknowledgements

We acknowledge the helpful comments of the late Dr. Norman L. Johnson.

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Authors and Affiliations

  1. Department of Environmental Sciences and Engineering, School of Public Health, University of North Carolina, CB 7431 Rosenau Hall, Chapel Hill, North Carolina, USA

    Michael R Flynn

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  1. Michael R Flynn
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Correspondence to Michael R Flynn.

Appendix A

Appendix A

SB equations:

The mean of y is (Johnson, 1949)

where φ=A−B and ψ=CD and

The variance of y is

where μ′2 is the second moment of y about the origin and

Application of the chain rule to expression (A.7) and some algebraic simplification yields the following explicit expression for the variance of y:

The partial derivatives are

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Flynn, M. Fitting human exposure data with the Johnson SB distribution. J Expo Sci Environ Epidemiol 16, 56–62 (2006). https://doi.org/10.1038/sj.jea.7500437

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