Probability Plot Goodness‐of‐Fit and Skewness Estimation Procedures for the Pearson Type 3 Distribution

Abstract

Uniform flood frequency guidelines in the United States currently recommend fitting a Pearson (P3) distribution to the logarithms of annual maximum flood flows. As a result, a plethora of procedures have been recommended for obtaining unbiased plotting positions and unbiased estimates of the skew coefficient and for inverting the cumulative distribution function of a P3 variate. These developments are precisely the ingredients required for the construction of P3 probability plots. Using Monte Carlo simulation, we develop a probability plot correlation coefficient (PPCC) hypothesis test for the P3 distribution. Power studies are performed to evaluate the ability of the test to discriminate among competing distributional alternatives and to enhance our understanding of why the P3 distribution often appears to provide such a good fit to observed flood flow data. A new estimator of the skew coefficient is presented which, unlike the biased and unbiased moment estimators, is unbounded and has significantly lower root mean square error than the moment estimators for highly skewed samples.

Cramer–von Mises and Anderson‐Darling goodness of fit tests for extreme value distributions with unknown parameters

“…The situation is reversed (A 2 does slightly better than PP) when the hypothetical distribution is EV2 or LP3, while for the EV1 and GAM distributions the advantage of using A 2 is substantial. The weakness of the PP test for the GAM distribution was recognized also by Vogel and McMartin [1991], and can be attributed to the difficulty of extending to three-parameter distributions a test statistic which arises naturally for distributions with only location and scale parameters.…”

“…This is probably due to a widespread distrust of classical goodness of fit tests, when applied to small samples, and to the lack of a valid alternative for the case when the parameters of the hypothesized distribution are estimated using the same sample that is being tested. Important testings techniques, borrowed from applied statistics, have been proposed in the hydrologic field by Vogel [1986] and Vogel and McMartin [1991], with an approach based on the probability plot correlation coefficient (PP tests), by Ahmad et al [1988], using empirical distribution function statistics (EDF tests), and by Chowdhury et al [1991], Fill and Stedinger [1995], and Wang [1998], with techniques based on the comparison of empirical and hypothetical L moments ratios (LM tests).…”

“…No such tables exist for the GEV with the three parameters estimated from the sample. For the GAM distribution the testing procedure is described by Vogel and McMartin [1991].…”

The Probability Density Evolution Method for Flood Frequency Analysis: A Case Study of the Nen River in China

“…(3) Conduct the numerical integration with respect to zj according to Equation (11). Thereby, the discrete PDF will be written as:…”

“…Besides, the probability plot correlation coefficient (PPCC) test which was a powerful and easy-to-use goodness of fit test in completing samples for the composite hypothesis of normality was proposed by Filliben [9]. Thereafter, the PPCC test was extended for studying kinds of probability distribution types [10][11][12]. With respect to the parameter estimation research for the selected parent distribution, several methods have been extensively used, such as method of moments (MOM), method of maximum likelihood (ML), and curve-fitting method.…”