Parameter Estimation

This module estimates parameters for both S-FFA and NS-FFA. In NS-FFA, parameter estimation also involves estimating regression coefficients for time-varying parameters.

The framework supports three estimation methods:

  1. L-moments
  2. Maximum Likelihood (MLE)
  3. Generalized Maximum Likelihood (GMLE)

Note: We adopt the GEV distribution convention from Coles (2001)1, where a positive shape parameter \(\kappa\) indicates a heavy tail. This differs from the convention used by some other sources.

L-Moments

The L-moments parameter estimation method is implemented for all distributions in S-FFA. This method uses the sample L-moments (\(l_1\), \(l_2\)) and L-moment ratios (\(t_3\), \(t_4\)) to estimate parameters. For more information about L-moments, see here.

Warning: L-moment-based estimates can yield distributions which do not have support at small values. However, this is typically not an issue for quantile estimation of mid- to high-return periods.

Maximum Likelihood (MLE)

MLE is implemented for all distributions across both S-FFA and NS-FFA.

Maximum likelihood estimation aims to maximize the log-likelihood function \(\ell(x : \theta)\) of the data \(x = x_{1}, \dots , x_{n}\) given the parameters \(\theta\). The log-likelihood functions for each distribution are defined here. To find the optimal parameters, we use the nlminb function from the stats library. This function implements the "L-BFGS-B" algorithm for box-constrained optimization.

Generalized Maximum Likelihood (GMLE)

GMLE is used for GEV models when incorporating prior knowledge2 of the shape parameter \(\kappa\) using Bayesian reasoning via maximum a posteriori estimation, which maximizes the product of the likelihood and the prior distribution.

Suppose that \(\kappa\) is drawn from \(K \sim \text{Beta}(p, q)\) where \(p\) and \(q\) are determined using prior knowledge. The prior PDF \(f_{K}(\kappa)\) is shown below, where \(B(p, q)\) is the Beta function.

\[ f_{K}(\kappa) = \frac{\kappa ^{p - 1}(1 - \kappa)^{q-1}}{B(p, q)} \]

As in the case of regular maximum likelihood estimation, the likelihood function is:

\[ f_{X}(x : \mu, \sigma, \kappa) =\prod_{i=1}^{n} \frac{1}{\sigma}t_{i}^{-1 - (1/\kappa)} \exp (-t_{i}^{-1/\kappa}), \quad t_{i} = 1 + \kappa \left(\frac{x_{i} - \mu }{\sigma } \right) \]

As mentioned previously, we want to maximize the product \(\mathcal{L} = f_{K}(\kappa)f_{X}(x:\mu ,\sigma ,\kappa)\). To ensure numerical stability, we will maximize \(\ln (\mathcal{L})\) instead, which has the following form:

\[ \begin{aligned} \ln(\mathcal{L}) &= \ln(f_{K}(\kappa)) + \ln(f_{X}(x:\mu ,\sigma ,\kappa )) \\[10pt] \ln(f_{K}(\kappa)) &= (p - 1)\ln \kappa + (q-1) \ln (1 - \kappa) - \ln (B(p, q)) \\[5pt] \ln(f_{X}(x:\mu ,\sigma ,\kappa )) &= \sum_{i=1}^{n} \left[-\ln \sigma - \left(1 + \frac{1}{\kappa }\right) \ln t_{i} - t_{i}^{-1/\kappa}\right] \end{aligned} \]

  1. Coles, S. (2001). An introduction to statistical modeling of extreme values. Springer. ↩

  2. Martins, E. S., and Stedinger, J. R. (2000). Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resources Research, 36(3), 737–744. \doi{10.1029/1999WR900330} ↩