Multi-distribution regula-falsi profile likelihood method for nonstationary hydrological frequency analysis

Abstract

The recently developed regula-falsi profile likelihood (RF-PL) method has potential for quantifying uncertainty in nonstationary hydrological frequency analysis. However, its applicability to diverse distributions is constrained and lacks comprehensive evaluation. This paper extends the RF-PL method to multiple distributions (Gumbel, Generalized Logistic (GLO), Generalized Normal (GNO), Log-Normal, Log-Pearson type III, and Pearson type III (PE3)) by introducing their reparametrized log-likelihood functions. The extended multi-distribution RF-PL (MD-RF-PL) method is systematically assessed in a simulation study and practical applications, covering diverse nonstationary scenarios, distributions, L-skewness (${\tau }_{3}$), and return periods, and compared with the bootstrap and/or the conventional PL as benchmarks. The findings indicate that the MD-RF-PL method is computationally comparable to the bootstrap method but overall superior in capturing the true quantiles with reasonably wide confidence intervals when ${\tau }_{3}$ = 0.1 (moderate tails) and 0.3 (heavy tails). However, when ${\tau }_{3}$ = 0.5 (very heavy tails), although the MD-RF-PL method is superior and roughly equivalent to the bootstrap method for GEV and GLO distributions, respectively, it is inferior for GNO and PE3 distributions, albeit avoiding the occasionally extremely wide confidence intervals of the bootstrap method. This low performance of the MD-RF-PL method is associated with numerical instability. Moreover, the MD-RF-PL method reduces the computational demand of the PL method by 94–99% without degrading its accuracy. Both the simulation study and the practical applications consistently support the preference of the MD-RF-PL method for distributions with moderate to heavy tails and highlight the need for improving its numerical stability for very heavy tails.

Appendices

Appendix 1: Estimation of confidence interval bounds using the RF-PL method

The RF-PL method considers the target function $f({y}_{T,{t}_{i}}) ={\mathcal{l}}_{p}\left({y}_{T,{t}_{i}}\right)-{\mathcal{l}}_{p}^{thr}$, where ${y}_{T,{t}_{i}}$ is the effective return level (i.e., the quantile corresponding to a particular $T$ at time ${t}_{i}$), ${\mathcal{l}}_{p}\left({y}_{T,{t}_{i}}\right)$ is the profile likelihood of ${y}_{T,{t}_{i}}$, and ${\mathcal{l}}_{p}^{thr}$ is the confidence threshold (see Sect. 2.2). The RF-PL method consists of the following steps:

• 1. Define an initial pair of points denoted by ${y}_{T,{t}_{i}}^{a}$ and ${y}_{T,{t}_{i}}^{b}$ at a given $T$ and ${t}_{i}$, which enclose a sufficiently large interval such that it contains a zero of $f({y}_{T,{t}_{i}})$.

• 2. Calculate the approximated zero as ${y}_{T,{t}_{i}}^{c}=\frac{{y}_{T,{t}_{i}}^{b}f\left({y}_{T,{t}_{i}}^{a}\right)-{y}_{T,{t}_{i}}^{a}f\left({y}_{T,{t}_{i}}^{b}\right)}{f\left({y}_{T,{t}_{i}}^{a}\right)-f\left({y}_{T,{t}_{i}}^{b}\right)}$.

• 3. Re-define the interval as $[{y}_{T,{t}_{i}}^{a},{y}_{T,{t}_{i}}^{c}]$ if $f\left({y}_{T,{t}_{i}}^{a}\right)f\left({y}_{T,{t}_{i}}^{c}\right)<0$ or as $[{y}_{T,{t}_{i}}^{c}, {y}_{T,{t}_{i}}^{b}]$ if $f\left({y}_{T,{t}_{i}}^{c}\right)f\left({y}_{T,{t}_{i}}^{b}\right)<0$, such that the interval keeps containing the zero.

• 4. Stop the iteration if the error $f\left({y}_{T,{t}_{i}}^{c}\right)$ is less than or equal to the pre-defined error tolerance $\varepsilon$; otherwise, repeat steps 2 and 3 until the error is less than or equal to $\varepsilon$.

• 5. Repeat steps 1 to 4 for all the $T$ s and ${t}_{i}$ s of interest.

The $\varepsilon =\left|0.01\left[\mathcal{l}\left(\widehat{{\varvec{\theta}}}\right)-{\mathcal{l}}_{p}^{thr}\left({\theta }_{j}\right)\right]\right|$ was adopted after verifying it was adequate for the target problem after scrutiny (Vidrio-Sahagún and He 2022a). The $\widehat{{\varvec{\theta}}}$ is obtained by maximizing ${\mathcal{l}}_{p}\left({\theta }_{j}\right)$ as in the maximum likelihood estimation method.

Appendix 2: Candidate distributions

See Table 3.

Table 3 Specifications of the six candidate distributions, including their cumulative distribution function (${F}_{Y}$), quantile function (${y}_{T}$) in terms of the return period $T=1/p$ (where $p$ is the exceedance probability), and support

Appendix 3: Nonstationary generic log-likelihood functions

See Table 4.

Table 4 Generic log-likelihood functions of the selected nonstationary distributions