The method of self-determined probability weighted moments revisited

Abstract

Haktanir originally introduced the method of self-determined probability weighted moments as an extension of the traditional method of probability weighted moments for parameter estimation. While this method possesses many advantages, his algorithms introduced certain mathematical manipulations for numerical convenience or based upon special knowledge of the behavior of a data sample. Also, some of these algorithms relied upon inputs from numerical tables that are not widely accessible. To improve the usefulness of this method, new algorithms have been developed that directly implement the relevant equations and do not rely upon external results. In this paper, we show that these features extend the applicability of self-determined probability weighted moments without loss of accuracy in the parameter estimates. Examples from flood peak analysis and extreme wind speed estimation are presented.

Introduction

The problem of estimating the parameters of a probability distribution from a sample is crucial to many fields of science and engineering, particularly for predicting future behavior of a phenomenon from previously observed behavior. A wide variety of methods have been developed to perform parameter estimation; see, e.g. Rao and Hamed, 2000 for a discussion of some commonly used methods. Despite the efforts of many researchers, there is an on-going need to create a method that is easily used, has the flexibility to accommodate many different distributions, and can produce accurate and robust parameter estimates from (frequently limited) sets of data.

The method of self-determined probability weighted moments (Haktanir, 1997) was originally introduced as an extension of the traditional method of probability weighted moments (Greenwood et al., 1979) for parameter estimation. One goal of the self-determined probability weighted moment (SD-PWM) method was to enhance the accuracy of the probability weighted moment (PWM) method by more fully utilizing the mathematical properties of the underlying probability distribution. (This could also provide an informal ‘test’ of the appropriateness of the assumed distribution in describing the data, as explained more fully in Section 2.) In addition, the SD-PWM method could more accurately account for variations in the data sample, thus improving the parameter estimation. Haktanir (1997) documented the improved performance of the SD-PWM method over the PWM method and the maximum likelihood method by estimating parameters from five distributions (generalized extreme-value (GEV), log-logistic (LL), three parameter lognormal (LN3), and Pearson type three (P3), and Gumbel) using annual flood data from a location in Turkey.

Because of the general complexity of the SD-PWM equations to be solved, Haktanir introduced numerical algorithms to determine the SD-PWM parameter estimates. While these algorithms are for the most part direct implementations of the SD-PWM equations, certain mathematical manipulations were introduced for numerical convenience or based upon special knowledge of the behavior of a data sample. Moreover, some of these algorithms relied upon multiple inputs from numerical tables that are published in journals that are not widely accessible. While not detracting from the accuracy of his results, all of these features make the algorithms harder to use and potentially susceptible to problems when applied to other types of data.

In an attempt to simplify and unify the enforcement of the definition of SD-PWMs across the various distributions, new SD-PWM algorithms have been developed for each distribution. These new algorithms are entirely self-contained and directly implement the theoretical equations for the SD-PWM method with no modifications. They rely upon powerful and efficient numerical techniques for integration and root finding implemented in the software package matlab^® (1999), thus assuring accuracy and wide accessibility. These new algorithms have been tested both on Haktanir's original flood data and on extreme wind speed data, and their performance is assessed below. In general, the new algorithms perform as well as Haktanir's original ones, and they are shown to eliminate certain problems that were found in the Haktanir's implementation. Thus, these new algorithms should be more appropriate for general use in parameter estimation while not sacrificing the quality of Haktanir's approach.

The organization of the paper is as follows. In Section 2, we describe the theory of both probability weighted moments and self-determined probability weighted moments, highlighting the advantages that the latter method should have over the former. Section 3 gives implementation details for three of the distributions and points out differences with Haktanir's approach. In Section 4, results of testing of the new algorithms are reported. Finally, conclusions and future directions of the research are discussed in Section 5.