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

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: $10\le T\le \mathrm{10,000}\mathrm{years}$ 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 ($=5\text{skewnesses}\times 3\text{distributions}$). 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 $1\mathrm{million}/n$ short series of lengths: $n=20$, 30, 50, 100, 200. Instead of biases and root mean square errors of $1\mathrm{million}/n$ 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 (${\mathrm{MRD}}_j$s), mean absolute relative differences (${\mathrm{MARD}}_j$s), standard deviations of relative differences (${\mathrm{SDRD}}_j$s), and standard deviations of absolute relative differences (${\mathrm{SDARD}}_j$s) of the areas between the frequency curves of the short series and the frequency curve of the parent distribution over the entire range: $0.0001\le \mathrm{Pex}\le 0.1$ are proposed. Ranked tables of ${\mathrm{MRD}}_j$s, ${\mathrm{MARD}}_j$s, ${\mathrm{SDRD}}_j$s, and ${\mathrm{SDRD}}_j$s computed from $1\mathrm{million}/n$ $n$-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.