Correcting confidence intervals for quantiles of a heavy-tailed distribution: Case of the two-parameter Kappa distribution

- We consider the 2-parameter Kappa distribution.
- We estimate quantiles of this distribution by maximum likelihood.
- We calculate confidence intervals for quantiles by a traditional method.
- We suggest an improved method for calculating confidence intervals for quantiles.

SummaryIn modeling hydrological phenomena, statistical distributions are commonly used as frequency models to fit hydrological data. The 2-parameter Kappa (KAP) distribution has been proposed to analyze precipitation, wind speed and stream-flow data. Estimates of distribution quantiles are important risk measures of the frequency of occurrence of extreme hydrological events, and the calculation of confidence intervals for these quantiles (CIQs) is also essential, as it provides a measure of the statistical error involved in the estimation. This study revisits the most frequently used method for calculating CIQs for the KAP distribution and proposes a method for improving their accuracy. The calculation of CIQs has traditionally been based on the large-sample assumption that the quantile estimators are normally distributed, but with small samples commonly available, this assumption is shown to be quite crude. It is shown that significantly more accurate CIQs are obtainable if the KAP quantile estimators are transformed to better fit a normal distribution, and then corrected for possible bias. The comparison among CIQs is done on the basis of their coverage probabilities of the true distribution quantile. The results of the comparison lead to improved methods for calculating CIQs for the KAP distribution, the application of which is illustrated through a hydrological example. Although the study restricts attention to the maximum likelihood (ML) fitting method, we anticipate that the drawn recommendations would apply to other fitting methods also.