Robust Local Likelihood Estimation for Non-stationary Flood Frequency Analysis

Abstract

As changes to the environment, both human and non-human made, continue to occur, the assumption of stationarity in flood frequency analysis is seldom met. A proposed method for estimating the 1% chance flood, i.e., Q$_{100}$ of a flood gauge’s annual peak streamflows, using robust local likelihood estimation is developed. Simulations indicate that when a flood series seems to be from a more mixed population of values, often due to extreme snowmelt or tropical storms in a given flood year, robust local likelihood estimation is more effective at estimating the 1% chance flood than local likelihood estimation. Annual peak streamflows from the Congaree River at Columbia, South Carolina, the Illinois River at Marseilles, Illinois, and the Winooski River at Montpelier, Vermont are used as examples on how to apply the robust local likelihood method.

Appendix

We need to derive and evaluate the Jacobian at $\left\{ t_i\right\}$ in order to construct a sandwich estimator for the covariance matrices of their respective maximum likelihood estimates. The density function $f(\cdot ;\cdot )$ for each component can be written as:

$$\begin{aligned}f\left( y_{i};{\varvec{\Theta }}_{t}\right) = \frac{1}{\tau _t}k_{1}\left( y_{i}\right) ^{\delta _{t}+1}e^{-k_{1}\left( y_{i}\right) } \end{aligned}$$

where

$$\begin{aligned}k_{1}\left( y_{i}\right) = \left[ 1+\delta _{t}\left( \frac{y_{i}-\eta _t}{\tau _t}\right) \right] ^{-1/\delta _{t}}. \end{aligned}$$

The log of the density can be written as:

$$\begin{aligned} \log f\left( y_{i}|{\varvec{\Theta }}_{t}\right) = -\log \tau _t +\left( \delta _{t}+1\right) \log k_{1}\left( y_{i}\right) -k_{1}\left( y_{i}\right) \end{aligned}$$

The key in the following calculations is $k_{1}$; of particular utility are the following results:

$$\begin{aligned} \frac{\partial }{\partial \beta _{0,t}}k_{1}\left( y_{i}\right)= & {} \frac{k_{1}\left( y_{i}\right) }{\tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) } \\ \frac{\partial }{\partial \beta _{1,t}}k_{1}\left( y_{i}\right)= & {} \frac{\left( i-nt\right) k_{1}\left( y_{i}\right) }{\tau _{it} +\delta _{t}\left( y_{i}-\eta _{it} \right) } \\ \frac{\partial }{\partial \gamma _{0,t}}k_{1}\left( y_{i}\right)= & {} \frac{\left( y_{i}-\eta _{it}\right) k_{1}\left( y_{i}\right) }{\tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) } \\ \frac{\partial }{\partial \gamma _{1,t}}k_{1}\left( y_{i}\right)= & {} \frac{\left( i-nt\right) \left( y_{i}-\eta _{it}\right) k_{1}\left( y_{i}\right) }{\tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) } \\ \frac{\partial }{\partial \delta _{t}}k_{1}\left( y_{i}\right)= & {} -\frac{k_{1}\left( y_{i}\right) }{\delta _t}\left\{ \log k_{1}\left( y_{i}\right) +\frac{y_{i}-\eta _{it}}{\left[ \tau _{it}+\delta _t\left( y_{i}-\eta _{it}\right) \right] }\right\} . \end{aligned}$$

Using the results derived for $k_1$ above, the components of the score vector are given below.

$$\begin{aligned} \frac{\partial }{\partial \beta _{0,t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{\delta _{t}+1-k_{1}\left( y_{i}\right) }{\tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) } \\ \frac{\partial }{\partial \beta _{1,t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{\left( i-nt\right) \left[ \delta _{t}+1-k_{1}\left( y_{i}\right) \right] }{\tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) } \\ \frac{\partial }{\partial \gamma _{0,t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} -1+ \frac{\left[ y_{i}-\eta _{it}\right] \left[ \delta _{t}+1-k_{1}\left( y_{i}\right) \right] }{\tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) } \\ \frac{\partial }{\partial \gamma _{1,t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} -\left( i-nt\right) + \frac{\left( i-nt\right) \left[ y_{i}-\eta _{it}\right] \left[ \delta _{t}+1-k_{1}\left( y_{i}\right) \right] }{\tau _{it}+\delta _{t}\left[ y_{i}-\eta _{it}\right] } \\ \frac{\partial }{\partial \delta _{t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \log k_{1}\left( y_{i}\right) - \frac{\delta _{t}+1-k_{1}\left( y_{i} \right) }{\delta _{t}} \left[ \log k_{1}\left( y_{i}\right) + \frac{y_{i}-\eta _{it}}{\tau _{it}+\delta _t \left( y_{i}-\eta _{it}\right) } \right] . \end{aligned}$$

These formulas, evaluated at $\left\{ t_i\right\}$, can be used to construct the score $\frac{\partial }{\partial {\varvec{\Theta }}_{i}}\log l({\varvec{\Theta }}_{i}) = {\varvec{s}}_i$ for each observation i; the total score can be computed as ${\varvec{S}}\left( {\varvec{\Theta }}\right) = \sum _{i=1}^n {\varvec{s}}_i$.

The components summed to comprise the elements of J(t) for $f_{1}\left( y_{i}|{\varvec{\Theta }}_{t}\right)$, the observed information matrix for the local linear model, can be written as:

$$\begin{aligned} \frac{\partial ^2}{\partial \beta ^2_{0,t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{\left( \delta _{t}+1\right) \left( \delta _{t}-k_{1}\left( y_{i}\right) \right) }{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \beta _{0,t}\partial \beta _{1,t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{\left( i-nt \right) \left( \delta _{t}+1\right) \left( \delta _{t}-k_{1}\left( y_{i}\right) \right) }{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \beta _{0,t}\partial \gamma _{0,t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{-\left( y_{i}-\eta _{it} \right) - \left( \delta _{t}+1-k_{1}\left( y_{i}\right) \right) \tau _{it}}{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \beta _{0,t}\partial \delta _{t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{\left( \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right) \left( 1 - k_1'\right) - \left( \delta _t + 1 - k_{1}\left( y_{i}\right) \right) \left( y_{i}-\eta _{it} \right) }{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \beta _{1,t}^2}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{\left( i-nt \right) ^2\left( \delta _{t}+1\right) \left( \delta _{t}-k_{1}\left( y_{i}\right) \right) }{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \beta _{1,t}\partial \gamma _{0,t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{-\left( i-nt \right) \left[ \left( y_{i}-\eta _{it} \right) + \left( \delta _{t}+1-k_{1}\left( y_{i}\right) \right) \tau _{it}\right] }{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \beta _{1,t}\partial \delta _{t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{\left( i-nt \right) \left\{ \left( \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right) \left( 1 - k_1' \right) - \left( \delta _t + 1 - k_{1}\left( y_{i}\right) \right) \left( y_{i}-\eta _{it} \right) \right\} }{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \gamma _{0,t}^2}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{k_{1}\left( y_{i}\right) \left( y_{i}-\eta _{it}\right) ^2 - \left( y_{i}-\eta _{it}\right) \left( \delta _t + 1 - k_{1}\left( y_{i}\right) \right) \tau _{it}}{\left[ \tau _{it} +\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \gamma _{0,t}\partial \delta _{t}}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \frac{\left( y_{i}-\eta _{it}\right) \left( \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right) \left( 1-k_1'\right) -\left( y_{i}-\eta _{it}\right) ^2\left( \delta _t + 1 - k_{1}\left( y_{i}\right) \right) }{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2} \\ \frac{\partial ^2}{\partial \delta _{t}^2}\log f\left( y_{i}|{\varvec{\Theta }}_{t}\right)= & {} \left[ \log k_{1}\left( y_{i}\right) + \frac{\left( y_{i}-\eta _{it}\right) }{\left( \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right) }\frac{\delta _t k_1' + 1 -k_{1}\left( y_{i}\right) }{\delta _t^2} \right] \\ {}+ & {} \frac{k_1'\left( 1-k_1\right) }{k_{1}}\left( 1 + \frac{k_1}{\delta _t} \right) +\frac{\delta _t + 1 - k_{1}\left( y_{i}\right) }{\delta _{t}} \frac{\left( y_{i}-\eta _{it}\right) ^2}{\left[ \tau _{it}+\delta _{t}\left( y_{i}-\eta _{it}\right) \right] ^2}, \\ \end{aligned}$$

where

$$\begin{aligned} k_1'=\frac{\partial }{\partial \delta _{t}}k_{1}\left( y_{i}\right) . \end{aligned}$$