Bootstrap is a computer-based resampling and statistical inference method that can provide an estimate for the uncertainty of
distribution parameters and quantiles in frequency analysis. In this article, in addition to calculating the bootstrap confidence
interval for quantiles with percentile bootstrap (BP), accelerated bias-corrected bootstrap (BCA), and t-bootstrap methods,
the calculation of the confidence interval is proposed using the bootstrap highest density method (HDI) for the probability
distributions used in hydrology data. We obtain the average length of the confidence interval as a criterion for evaluating the
methods.
To calculate the average length of the confidence interval with different methods, first, the best distribution among widely
used distributions is fitted to the original data, and the parameters of the fitted distribution are estimated using the maximum
likelihood method to obtain the quantiles. Then, we continue by repeating the simulated bootstrap samples until the probability of covering the real quantile reaches the nominal confidence level of 0.95. The simulation results show that the bootstrap
highest density method gives the lowest average confidence interval length among all methods.
In previous studies, the probability of coverage with the same number of samples as the original sample and the number
of bootstrap repetitions (for example, 1000) have been obtained, and finally, the coverage probability closest to the nominal
value of 0.95 was chosen as the optimal state. In this article, to avoid a large number of bootstrap iterations, considering
the number of different samples n = 20, 40, 60, . . . , 160 that are simulated from the mother distribution, we continue the
number of iterations of the bootstrap samples only until reaching the coverage probability of 0.95. Usually, two-parameter
distributions require fewer samples than three-parameter distributions. In terms of the number of necessary repetitions (B)
to reach the 95% confidence level, the t-bootstrap method usually required fewer repetitions, although the implementation
time of this method was longer in R software.
The best distribution for the data used in this research is the two-parameter Frechet distribution, which was selected as
the best distribution in 80% of the stations, and can be used for similar studies dealing with maximum annual precipitation
values. According to the fitted distributions for the data of different stations, two-parameter distributions always fit the data
better than three-parameter distributions.
In general, the widest confidence intervals were obtained with BCA and t-bootstrap metho