Exploring generalized probability weighted moments, generalized moments and maximum likelihood estimating methods in two-parameter Weibull model
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
The quantities Mi,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 havewhere 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).
Section snippets
GPWM estimators in the two-parameter Weibull distribution
In hydrology, the three-parameter WEI distribution was one of the main models considered by Matalas, 1963, Loganathan et al., 1985, among others, as suitable for low stream-flow analysis. In the peaks-over-threshold (POT) method for modeling hydrological extremes, the generalized Pareto, two-parameter WEI, and exponential distributions have been widely used for fitting peaks over the threshold, or ‘exceedances’ (Rasmussen et al., 1993, Ouarda and Ashkar, 1995). These three distributions were
Application
The knowledge of various flood characteristics in river channels (duration, volume, etc.) is essential for water management and the design of hydraulics structures. We consider here, the volume of flood events for hydrometric Station BP001, on the Little Southwest Miramichi River, at Lyttleton, in New Brunswick, Canada. The drainage area for this station is 1340 km2. The series from which flood volumes were calculated is composed of 45 years of daily river-flows (1952–1996). We were interested
Summary
After comparing it to two other estimation methods, GPWM and GM, the ML method was the one generally recommended in the present study for fitting the two-parameter WEI model. However, the GPWM or GM methods may perform better for other distributions, when the sample size is small or moderate, as has been shown by some investigators. On the other hand, one added advantage for using ML to fit the two-parameter WEI model, is that a small-sample procedure is available for calculating confidence
Acknowledgements
The financial support of UWI and of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. The authors wish also to thank Miss Lampouguin Bayentin for her assistance in preparing Section 3.
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