Parameter Estimation for Univariate Hydrological Distribution Using Improved Bootstrap with Small Samples

It is crucial yet challenging to estimate the parameters of hydrological distribution for hydrological frequency analysis when small samples are available. This paper proposes an improved Bootstrap and combines it with three commonly used parameter estimation methods, i.e., improved Bootstrap with method of moments (IBMOM), maximum likelihood estimation (IBMLE) and maximum entropy principle (IBMEP). A series of numerical experiments with different small sized (10, 20, and 30) of samples generated from the three commonly used probability distributions, i.e., Pearson Type III, Weibull, and Beta distributions, are conducted to evaluate the performance of the proposed three methods compared with the cases of conventional Bootstrap and without-Bootstrap. The proposed methods are then applied to the estimation of distribution parameters for the average annual precipitations of 8 counties in Qingyang City, China with assumption of Pearson Type III distribution for the average annual precipitations. The resulting absolute deviation (AD) box plots and Root Mean Square Error (RMSE) and bias estimators from both the numerical experiments and the case study show that the estimated parameters obtained by the improved Bootstrap methods have less deviation and are more accurate than those obtained through conventional Bootstrap and without-Bootstrap for the three distributions. It is also interestingly found that the improved Bootstrap provides more relative improvement on the parameter estimation when smaller size of sample is used. The method based on improved Bootstrap paves a new way forward to alleviating the need of large size of sample for quality hydrological frequency analysis.

Automated summary

This study proposes an improved Bootstrap method and combines it with three commonly used parameter estimation methods, i.e., improved Bootstrap with method of moments (IBMOM), maximum likelihood estimation (IBMLE), and maximum entropy principle (IBMEP). A series of numerical experiments and a case study on the estimation of distribution parameters demonstrate that the proposed methods provide more accurate and less deviated results compared to conventional Bootstrap and without-Bootstrap approaches. Moreover, the improved Bootstrap method shows significant improvement in parameter estimation when smaller sample size is used. The method based on improved Bootstrap offers a new solution to the requirement of large sample size in quality hydrological frequency analysis.