Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 11 2 2026 09 01 Bootstrap Highest Density Confidence Interval by Comparing Two Climatological Regions 324 332 2132 10.22128/ansne.2026.3310.1214 EN Ali Shadrokh Department of Statistics, Payame Noor University, P.O.Box 19395-4697, Tehran, Iran 0000-0003-0069-8165 Mehdi Pejman Department of Statistics, Payame Noor University, P.O.Box 19395-4697, Tehran, Iran Journal Article 2026 05 06 Bootstrap is a resampling method based on high calculations, which can help us a lot for statistical inference in cases where the amount of data that we have is limited. For example, in the design of hydraulic structures such as bridges or dams, etc. there is a need to estimate hydrological events such as, floods or precipitations by statistical inference of quantiles of a probability distribution. In this paper, we aim to estimate precipitation quantiles. For calculating this estimation, the confidence interval for quantiles has been introduced with percentile bootstrap, accelerated bias-corrected bootstrap, t-bootstrap methods; that in this article, we want to compare these methods with the confidence interval made by the highest density method based on bootstrap data and we obtain the average length of the confidence intervals as a criterion to evaluate the methods. To calculate the average length of confidence intervals using different methods, first, the best distribution among commonly used distributions is fitted to the original data, and its parameters are estimated using the maximum likelihood method, and quantiles are obtained from it. Then, we continue until the coverage probability of the real quantile reaches the nominal confidence level of 95$\%$ by repeating the simulated bootstrap samples. The results of the performed simulation show that the bootstrap highest density method has the smallest average length of confidence intervals among all methods. The data used in the article are 24-hour annual maximum precipitation records in five meteorological stations in Mexico, which are compared with the data of five stations in Gilan province of Iran. https://ansne.du.ac.ir/article_2132_81dc48da06120a6111f5afe7cde2fdb7.pdf