## 1. Introduction

[2] The development of stochastic methods for the characterization of flood peaks in drainage basins has both motivated and benefited from the treatment of classical problems in extreme value statistics. The generalized extreme value (GEV) distribution has been widely used for modeling the distribution of flood peaks in at-site and regional settings [*Hosking et al.*, 1985a; *Smith*, 1987; *Stedinger and Lu*, 1995; *Rosbjerg and Madsen*, 1995; *Hosking and Wallis*, 1996]. In addition to flood modeling, the GEV distribution is commonly used to model many other natural extreme events [*Smith*, 1986; *Bauer*, 1996; *Kuchenhoff and Themerus*, 1996; *Bruun and Tawn*, 1998; *Parret*, 1998]. “Extreme events” are often defined to be the maximum value of a quantity over a given period of time, such as the maximum annual discharge in a river. Extreme value theory, in particular the extremal types theorem [*Leadbetter et al.*, 1983], suggests that the distribution of these maxima should be close to one of the extreme value types. The GEV distribution, introduced by *Jenkinson* [1955], is a three-parameter distribution that combines all three extreme value types into a single form (see section 2).

[3] Parameter estimation procedures for the three-parameter GEV distribution have been extensively studied (see the work of *Martins and Stedinger* [2000] for a literature review). The most commonly used methods are the maximum likelihood estimation (MLE) [*Prescott and Walden*, 1980], the method of L moments (LMOM) [*Hosking*, 1990], and the method of moments (MOM). It has been noted that estimates of the shape parameter *k* of the GEV distribution for flood peak data are usually negative [*Smith*, 1987; *Madsen et al.*, 1997; *Martins and Stedinger*, 2000], implying heavy tails in the distribution. It has been shown that MLE parameter estimators have a very large variance for negative values of *k*, and result in large errors in quantile estimation. Although both LMOM and MOM estimators tend to produce biased estimates, they are still considered preferable to MLE because of smaller variances in their quantile estimates [*Hosking et al.*, 1985b; *Madsen et al.*, 1997]. MLE-based methods, however, can easily incorporate additional information, such as censored data [*Prescott and Walden*, 1983] or a known prior distribution for *k* [*Martins and Stedinger*, 2000, 2001].

[4] In this paper (section 3), we show how MLE and LMOM methods can be combined to produce improved GEV parameter estimators based on annual maximum series (AMS) of flood peaks. The resulting “mixed” method estimators of the shape parameter of the GEV distribution have reduced variance compared to the MLE estimator and reduced bias compared to the LMOM estimator. The root mean square errors (RMSE) of mixed method estimators of the GEV location, scale, and shape parameters are superior to those of LMOM estimators for flood-size samples. The RMSE of LMOM estimators of extreme flood quantiles (100 and 1000 year flood magnitude analyses are presented) are slightly smaller than those for mixed method estimators. The contrasting properties of quantile and parameter estimators are examined and provide interesting insights to both LMOM and mixed method estimators.

[5] Partial duration series (PDS) [*Shane and Lynn*, 1964; *Todorovich and Zelenhasic*, 1970] models assume that the arrival times of peaks above a specified threshold form a Poisson process in time, and that the distribution of the peak magnitudes has a particular form. The attraction of these procedures is that additional information can be used, relative to AMS-based techniques. There are, for example, many flood records in which the second largest flood peak during a year is larger than the majority of other flood peaks. The generalized Pareto (GP) distribution is a common choice for the peak magnitude distribution both because it corresponds to a limiting distribution for excesses over a threshold as that threshold is increased [*Leadbetter et al.*, 1983], and the resulting distribution of annual flood peaks is GEV [*Smith*, 1984; *Madsen and Rosbjerg*, 1997]. *Madsen et al.* [1997] showed that errors in parameter estimation under the GP/PDS approach under certain conditions are smaller than those of the GEV/AMS approach.

[6] In section 4 we introduce a MLE method that uses the values of the two largest observations for a given year (MLE2) and extend the method to a mixed method estimator. The MLE2 method is developed in the GP/PDS framework and tested via Monte Carlo simulations. Analyses illustrate the flexibility of the mixed method framework and the potential for improving parameter and quantile estimators through incorporation of flood observations from PDS records.

[7] The GEV distribution has played an important role in regional flood frequency analyses [*Hosking et al.*, 1985a; *Lettenmaier et al.*, 1987; *Chowdhury et al.*, 1991; *Stedinger and Lu*, 1995; *Hosking and Wallis*, 1996]. A commonly used foundation for regional flood frequency analyses is the simple scaling theory, which assumes that appropriately scaled annual flood peaks have the same distribution in a hydrologically homogeneous geographical region. In the GEV approach, this means that the shape parameter *k* of the GEV distribution and the ratio of scale and location parameters are constant for all basins in the region.

[8] In the work of *Smith* [1992], a sample of 104 basin from the central Appalachian region was studied, and the hypothesis that simple scaling theory holds for this sample was rejected. In section 6 we further investigate the applicability of the index-flood theory to the central Appalachian region. In particular, we estimate the parameters of the GEV distribution for the same sample of basins used by *Smith* [1992] and *Hosking and Wallis* [1996] and study the dependence of parameter estimates on morphological and land cover characteristics of the basins.