Assessment of Right-Tail Prediction Ability of Some Distributions by Monte Carlo Analyses


JOURNAL OF HYDROLOGIC ENGINEERING, vol.18, no.5, pp.499-517, 2013 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 18 Issue: 5
  • Publication Date: 2013
  • Doi Number: 10.1061/(asce)he.1943-5584.0000620
  • Page Numbers: pp.499-517


The probability distributions of Gumbel, three-parameter lognormal (LN3), general extreme values (GEV), three-parameter gamma (G3), and three-parameter log-gamma (LG3), whose parameters are computed by the methods of moments (MOM), maximum-likelihood (ML), probability-weighted moments (PWM), and self-determined probability-weighted moments (SDPWM) are compared from the aspect of predicting the right-tail quantiles of return periods in the range: 10 <= T <= 10,000 years from finite-length sample series by a Monte Carlo analysis. The parameters of the LN3 distribution are also computed by the method of zero-sample-skewness. Synthetic series of 1 million elements having skewness coefficients: +0.5, +1, +2, +3, +5 are generated by LN3, GEV, and G3 distributions, separately, resulting in 15 different 1 million-element synthetic series (=5 skewnesses x 3 distributions). The right-tail quantiles having exceedance probabilities (Pex) 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001, 0.0005, 0.0002, 0.0001 are first computed by the parent distribution. The right-tail quantiles having those Pexs are also computed by these 21 probability models using all 1 million/n short series of lengths: n = 20, 30, 50, 100, 200. Instead of biases and root mean square errors of 1 million/n differences of quantiles from those of the parent distribution separately for individual return periods (T), (e. g., 100 years, 1,000 years), which has been the usual procedure so far, mean relative differences (MRD(j)s), mean absolute relative differences (MARD(j)s), standard deviations of relative differences (SDRD(j)s), and standard deviations of absolute relative differences (SDARD(j)s) of the areas between the frequency curves of the short series and the frequency curve of the parent distribution over the entire range: 0.0001 = Pex = 0.1 are proposed. Ranked tables of MRDjs, MARDjs, SDRDjs, and SDRDjs computed from 1 million/n n-element series are investigated as a more comprehensive criterion of goodness of a probability distribution to predict right-tail population quantiles from short-length sample series. The G3-PWM distribution is found to be better, followed by the LN3-MOM, LN3-PWM, G3-MOM, GEV-MOM, and LN3-ML distributions for the ranges covered. DOI: 10.1061/(ASCE)HE.19435584.0000620. (C) 2013 American Society of Civil Engineers.