Model Checking for GEVD

The model fit of GEVD measure after estimating the parameters by utilising residual plots function as defined by Equation (6),

$$res = \begin{cases} \left(1 + \frac{\gamma}{\sigma}(\mathbf{x} - \mu)\right)^{-1/\gamma} & \text{if } \gamma = 0\\ \exp\left[-\exp\left(-\frac{\mathbf{x} - \mu}{\sigma}\right)\right] & \text{if } \gamma \neq 0 \end{cases} \tag{6}$$

Ascertain by Reference [20] conversion of data to unit exponential distributed residuals is on the null assumption that GEVD fits the data.

#### *2.6. Return Period or Level Estimates*

The frequency of extreme quantiles incidence estimated with a fixed value of return level. The return level is the mean number of events taking place within a unit period, e.g., one year [71]. Return levels are essential for prediction purposes and estimated from stationary models. The expected return time is the number of time (years) one is expected to wait on average before the observation of another extreme event of at least the same intensity. If a threshold exceedance of a given probability of an observed extreme incidence in any given time (year) is *p*, then the mean return period *T* is such that *T* = 1/*p*.

#### *2.7. Test for Stationarity and Seasonality*

The stationarity of the data conducted by the augmented Dickey-Fuller (ADF) stationarity test on the assumption that there is no trend [72]. The quality of convergence of the weather extremes is access using the Kolmogorov–Smirnov (K-S) and Anderson–Darling goodness-of-fit tests. The K-S test, relying on the empirical study of the cumulative distribution function, is used to determine whether the sample is from the hypothesised continuous distribution. The K-S approach is less sensitive for normal distribution [72]. The Anderson-Darling test, an enhancement of the K-S test, compares the fit to the expected cumulative distribution function of the observed cumulative distribution function. This test gives more substantial weight to the tail of the distribution than the K-S test [72].

The assumption is that the data is from a population which is independent identically distribution (i.i.d). The alternative hypothesis is a two-tail test on the assumption that the data follow a monotonic trend. Thus, the following test statistics by Mann-Kendall determine by Equation (7):

$$S = \sum\_{k}^{n-1} \sum\_{j=k+1}^{n} \text{sgn}\left(\mathbf{x}\_{j} - \mathbf{x}\_{k}\right) \tag{7}$$

with sgn the signum function.
