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Improved approximate confidence intervals for the mean of a log‐normal random variable
Author(s) -
Taylor Douglas J.,
Kupper Lawrence L.,
Muller Keith E.
Publication year - 2002
Publication title -
statistics in medicine
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.996
H-Index - 183
eISSN - 1097-0258
pISSN - 0277-6715
DOI - 10.1002/sim.1052
Subject(s) - confidence interval , range (aeronautics) , statistics , mathematics , numerical integration , random variable , sample size determination , variable (mathematics) , population , alpha (finance) , algorithm , sampling (signal processing) , simple (philosophy) , coverage probability , computer science , mathematical analysis , construct validity , materials science , demography , philosophy , filter (signal processing) , epistemology , sociology , composite material , computer vision , psychometrics
Data analysts often compute approximate 100(1‐α) per cent confidence intervals for the mean of a log‐normal random variable due to the computational effort required for exact intervals. We evaluate two simple approximations and demonstrate that the probabilities with which the intervals fail to capture the population mean (that is, the coverage error) can range from well above the desired level, α, to very near zero in small to moderate sample sizes ( n ⩽100). The performance of a more sophisticated approximation, implemented via numerical integration or bootstrap sampling, is noticeably improved, but also suffers from coverage errors that are too large when n ⩽25. A new procedure is developed which outperforms existing approximations. Computing these improved intervals requires the integration of standard distribution functions. The calculations are straightforward, however, and lead to satisfactory coverage errors for n as small as 5. A related method that avoids the integration step generally outperforms existing simple approximations for n ⩽100, while maintaining the coverage error at or below α. Programs to implement the new procedures are provided in an Appendix. Copyright © 2002 John Wiley & Sons, Ltd.