*2.2. Estimation of Parameters of AMS Probability Distribution*

Various distributions have been proposed for modeling extreme events, including the Generalized Extreme Value distribution (GEV), Generalized Pareto distribution (GP), gamma distribution, lognormal distribution, among others. The GEV distribution was recommended for flood frequency analysis in the United Kingdom (UK) Flood Studies Report [28]. According to the gauge-based Precipitation-Frequency Atlas of the US National Oceanic and Atmospheric Administration (NOAA) Atlas 14 [18], the GEV distribution provided an acceptable fit to data more frequently than any other distribution and was chosen to model the annual maximum series of all the stations covering the US southeastern states (Alabama, Arkansas, Florida, Georgia, Louisiana and Mississippi). These conclusions were obtained using a goodness-of-fit test based on L-moment statistics for 3-parameter distributions along with the results of χ<sup>2</sup> and Kolmogorov-Smirnov tests and visual inspection of probability plots. Naghavi & Yu [15] examined six different distributions for extreme precipitation over Louisiana and concluded that GEV distribution outperforms other distributions. Therefore, the GEV will be adopted in the current study to represent the AMS. The GEV distribution is a three-parameter distribution developed within the extreme value theory and combines three different models: Gumbel, Frechet and Weibull distributions, which are often referred to as Types (I), (II), and (III) distributions, respectively. The probability density function of the GEV distribution, in terms of its three parameters: Location (α), Scale (β), and Shape (κ -0), can be formulated as follows:

$$\mathbf{f}\_{\mathbf{X}}(\mathbf{x}) = \frac{1}{\beta} [1 - \frac{\kappa}{\beta}(\mathbf{x} - \alpha)]^{(1/\kappa - 1)} \mathbf{F}\_{\mathbf{X}}(\mathbf{x}) \tag{1}$$

In this study, the method of linear moments is used for the estimation of the GEV distribution parameters. The method of L-moments offers several advantages over other methods (e.g., method of moments, and method of maximum likelihood), especially in the cases of small sample sizes [29,30]. The L-moment estimators for the GEV distribution parameters are given as follows:

$$\text{\textquotedblleft} = 7.8590\text{c} + 2.9554\text{c}^2\text{\textquotedblright} \tag{2}$$

$$\text{lc.} = \frac{2}{\left(3 + \text{ ft.}\right)} - \frac{\log(2)}{\log(3)}\tag{3}$$

$$\hat{\mathfrak{a}}\_{-}=\hat{\lambda}\_{1}-\frac{\hat{\mathfrak{G}}}{\hat{\mathfrak{k}}}[1-\Gamma(1+\hat{\mathfrak{k}}\_{1})]\tag{4}$$

$$\text{(\ $\, \,)}\, \, \, \frac{\$ \, \, \, \, \&}{(1 - 2^{-k} \, \, \, k) \Gamma(1 + \, \, \&)}\tag{5}$$

where τ<sup>3</sup> (or L-skewness) is the ratio of the third and second L-moment (a measure of skewness), λ<sup>1</sup> is the first L-moment (measure of distribution mean), and λ<sup>2</sup> is the second L-moment (measure of the scale or dispersion). Accordingly, the quantiles corresponding to different return periods, T, (e.g., T = 5, 10, 50, 100 years) can be estimated as follows:

$$\mathfrak{x}(q) := \alpha + \frac{\hat{\mathfrak{k}}}{\mathfrak{k}} \left[ 1 - \left( -\ln q \right)^{\mathbb{A}} \right] \tag{6}$$

where *q* is the cumulative probability of interest that can be related to the return period T as - <sup>q</sup> = <sup>1</sup> <sup>−</sup> <sup>1</sup> T .
