Exploring generalized probability weighted moments, generalized moments and maximum likelihood estimating methods in two-parameter Weibull model

Abstract

Generalized probability weighted moments (GPWM), generalized moments and maximum likelihood (ML) estimating methods are investigated in the two-parameter Weibull (WEI) model. Point estimators for positive and negative shape parameters and for quantiles with special return periods are derived. Analytical expressions for the asymptotic variances of the estimators are presented. Simulation results on the performance of the three estimating methods are also given. The results show that the GPWM method may in some situations lead to a slight gain in quantile estimation accuracy. However, the overall results show the ML method to be the most recommendable one, since for the cases considered it performed either better or almost as good as the GPWM method. The WEI model is then used to fit a hydrological data set of flood volumes above a threshold, using data from the Little Southwest Miramichi River, in New Brunswick, Canada. It is shown that one added advantage for using ML method to fit the two-parameter WEI model is that small-sample procedures are available for calculating confidence intervals for WEI quantiles, when this method is used. It is recommended that such small-sample methods be used whenever available, in hydrology, for estimating distribution quantiles.

Introduction

The probability weighted moments (PWM) method, initiated by Greenwood et al. (1979), constitutes a leading alternative to the classical moments method (MM), and maximum likelihood (ML) method, for fitting statistical distributions to data. The ML method is the most important method since it leads to efficient parameter estimators with Gaussian asymptotic distributions. However, this method is often highly computational, whereas the MM method is mostly used because of its relative ease of application. Furthermore, the MM method can help to obtain starting values for numerical procedures involved in ML estimations. Since the ML method does not always work well in small samples, other estimating methods have recently been developed, as alternatives. Among these, the PWM method has been advocated by Hosking et al. (1985). The PWM of order (i,j,k) for a random variable X with cumulative distribution function (CDF) F and probability density function (PDF) f is defined as

$$\text{M}{}_{\mathrm{i,j,k}}\text{=E(X}{}^{\mathrm{i}}\text{[F(X)]}{}^{\mathrm{j}}\text{[1−F(X)]}{}^{\mathrm{k}}\text{)}$$

The quantities M_i,j,k exist if and only if E(Xⁱ) exists since |[F(X)]^j[1−F(X)]^k|≤1 for any positive integers j and k. From Eq. (1), we have

$$\text{M}{}_{\mathrm{i,j,k}}\text{=}\text{∫}\text{−∞}\text{∞}\text{x}{}^{\mathrm{i}}\text{(F(x))}{}^{\mathrm{j}}\text{(1−F(x))}{}^{\mathrm{k}}\text{f(x)}\text{d}\text{x=}\text{∫}\text{0}\text{1}\text{(x(F))}{}^{\mathrm{i}}\text{F}{}^{\mathrm{j}}\text{(1−F)}{}^{\mathrm{k}}\text{d}\text{F}$$

where x(F) is solution of the equation F(x)=F.

The order i=1 is the most commonly used, while the other orders, j and k, are often restricted to small positive integer values. The choice of i=1 has the double advantage of not over-weighting sample values unduly, and also leads to a class of linear L-moments (Hosking, 1986, Hosking, 1990) with asymptotic normality. We shall therefore concentrate on PWMs of order (1,j,k). Although only small positive integers are required to estimate the parameters of distributions, there is a lot to gain in using real numbers and not necessarily small ones, according to Rasmussen (2001). The extended class of PWMs with real orders is referred to by Rasmussen (2001) as the class of generalized probability weighted moments (GPWMs). However, GPWMs may not exist for some negative real numbers. The effect of using GPWMs for estimating the parameters and quantiles of a generalized Pareto distribution has been investigated by Rasmussen (2001). In this paper, we compare the accuracy of estimates of the shape parameter and quantiles of a two-parameter Weibull (WEI) distribution. GPWM, generalized moments (GM) and ML methods are considered. The GM method has been initiated and detailed by Ashkar and Bobée, 1987, Bobée and Ashkar, 1988, Bobée and Ashkar, 1991. It has been successfully applied for fitting the generalized Pareto distribution by Ashkar and Ouarda (1996).