Plotting Positions for the Generalized Extreme Value Distribution: A Critique

Abstract

Plotting ordered ranked data is a standard graphical technique applied in many hydrologic and water resource fields that utilizes the plotting position formula. This graphical approach mainly deals with assigning a probability of occurrence to extreme events like floods, rainfall, etc. The concept of unbiasedness has inspired several researchers to develop various plotting positions, including the shape parameter of a probability distribution. The derivation of Generalized Extreme Value (GEV) distribution form the statistical theory of extreme random variables justifies its vast application in the flood frequency studies. Considering the significance of GEV distribution in the field of hydrology, in the present study, existing plotting positions for GEV are compared in terms of root mean square error (RMSE) and relative bias (RBIAS) between theoretical reduced variates of GEV distribution and those obtained from these formulas. The Sum of square error between the top three reduced variates is used as a statistical indicator to evaluate their performance in the estimation of higher quantiles. The study is carried out varying the sample size (n) over the most frequently occurred range of shape parameter (k) in hydrological applications, i.e., −0.3 ≤ k ≤ 1.5. No single plotting position is found to perform better over the entire range of shape parameter. However, the formula considering the skewness coefficient of the sample satisfies comparatively a broad range of shape parameter. The necessity of modifying these plotting positions has been discussed to address the effect of sample size and skewness coefficient.KeywordsPlotting position formulaGEV distributionFlood frequencyShape parameterCoefficient of skewness