Development of plotting position for the general extreme value distribution

Probability plotting positions are used to graphically display the annual maximum rainfall or flood and to estimate the exceedance probabilities of those values. Therefore, the graphical approach using plotting positions has been applied in many hydrology and water resource engineering fields. The definition of unbiased plotting positions by Cunnane (1978) as the mean of the order statistics from reduced variates has influenced researchers to develop the plotting position of the probability distribution containing shape parameters. In this study, the plotting position formula for the general extreme value (GEV) distribution was derived by using the theoretical reduced variates of the GEV distribution for various sample sizes and shape parameters. To choose an appropriate plotting position formula, we examined eight plotting position formula types containing coefficients of skewness or squared coefficients of skewness in the numerator and/or denominator. In addition, the parameters of the plotting position formula for the GEV distribution were estimated by using a genetic optimization method known as the real-coded genetic algorithm (RGA). The accuracy of the derived plotting position formula for the GEV distribution was examined on the basis of the root mean square errors and relative bias between the theoretical reduced variates and those calculated from the derived and existing plotting position formulas. The derived plotting formula was found to be useful if the range of the shape parameter was within ±0.2.