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) from
obtained from the nonparametric method. They also allow estimates of extreme quantiles (e.g., y ˆ 0.99
small data sets (n < 20). This estimation involves extrapolation beyond the range of the observed values.
The danger in this extrapolation is in assuming the wrong population distribution.
The 50th percentile can be estimated with greater precision than any other can, and precision decreases
rapidly as the estimates move toward the extreme tails of the distribution. Neither estimation method
produces very precise estimates of extreme percentiles, even with large data sets.

References

Berthouex, P. M. and I. Hau (1991). “Difficulties in Using Water Quality Standards Based on Extreme
Percentiles,” Res. Jour. Water Pollution Control Fed., 63(5), 873–879.
Bisgaard, S. and W. G. Hunter (1986). “Studies in Quality Improvement: Designing Environmental Regula-
tions,” Tech. Report No. 7, Center for Quality and Productivity Improvement, University of Wisconsin–
Madison.
Crabtree, R. W., I. D. Cluckie, and C. F. Forster (1987). “Percentile Estimation for Water Quality Data,” Water
Res., 23, 583–590.
Gilbert, R. O. (1987). Statistical Methods for Environmental Pollution Monitoring, New York, Van Nostrand
Reinhold.
Hahn, G. J. and S. S. Shapiro (1967). Statistical Methods for Engineers, New York, John Wiley.

Exercises

8.1 Log Transformations. The log-transformed values of n = 90 concentration measurements have
an average value of 0.9 and a standard deviation of 0.8. Estimate the 99th percentile and its
upper 95% confidence limit.
8.2 Percentile Estimation. The ten largest-ranked observations from a sample of n = 365 daily
observations are 61, 62, 63, 66, 71, 73, 76, 78, 385, and 565. Estimate the 99th percentile
and its two-sided 95% confidence interval by the nonparametric method.
8.3 Highway TPH Data. Estimate the 95th percentile and its upper 95% confidence limit for the
highway TPH data in Exercise 3.6. Use the averages of the duplicated measurements for a
total of n = 30 observations.

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