Assessing the applicability of fractional order statistics for computing confidence intervals for extreme quantiles

SummaryThe main aim of frequency analysis of hydrological data is the definition of the quantiles with a given return period, also known as return levels, namely, the values xTxT of a hydrological variable X exceeded on average once in T time intervals. To make the analysis effective, these values have to be complemented with information about their uncertainty, which is usually quantified by confidence intervals (CIs). The paper discusses and compares some parametric, non-parametric, and simulation techniques to build CIs for quantiles with emphasis on the extreme ones. In particular, the work focuses on the usefulness of fractional order statistics to compute CIs for quantiles. This approach appears attractive as it involves simple analytical formulas and allows computing fully non-parametric CIs without requiring information about the parent distribution of the data. Since CIs for order statistics can be used to approximate CIs for quantiles based on order statistics, the relation between them is explored as well. The applicability of methods based on order statistics is assessed by studying the actual coverage probability through simulations, and applying these approaches to annual maximum peak discharge records from 22 streamgage stations in the continental United States and a streamgage station downtown Rome, Italy. For extreme quantiles, results show that these techniques are generally outperformed in term of coverage probability by other methods, such as Monte Carlo approach, when the parent distribution is well-specified or mis-specification involves distributions with tail behavior similar to the one used to generate the sample. On the other hand, the application to real world data indicates that the values of confidence limits based on order statistics are close to those obtained by Monte Carlo approach. In particular, the upper limits from non-parametric setups often result in cautionary values.