**6. Distribution of the exceedances**

the plot the MLE estimates for the parameters of the GPD against their 80 corresponding thresholds, within the range 0≤*v*≤250, 000, together with a 95%

the whole dataset (when the threshold,*v*, is zero). The re-parametrized scale parameter estimates on the other hand, seem to be stable beyond 50,000 but not so

The Gertensgarbe plot provides a more powerful procedure for threshold

We can tell that the estimates of the shape parameter appear to be constant for

**Statistic Value** *k*<sup>0</sup> 19 p-value 0.00014 threshold 150000 tail index 0.48507

The information provided by these plots can however, be rather approximative.

The cross point of the Gertensgarbe graph (**Figure 9**) is at the observation numbered *k* ¼ 19, which corresponds to a threshold of 150,000. The null hypothesis

confidence intervals.

*Mann-Kendall Test Results.*

*Natural Hazards - Impacts, Adjustments and Resilience*

**Table 2.**

beyond 225,000.

estimation [24].

**Figure 10.**

**288**

*Q-Q Plot of GPD under different thresholds.*

Given a threshold of 50,000, we are interested in the distribution of the excesses. The number of exceedances is 22, and **Figure 11** shows the plot of the kernel density estimates of the number and value of the exceedances.

The number of exceedances is assumed to be Negative-binomial-distributed. We carry out a chi-squared test using the MLE estimates to test its goodness of fit:

**Table 3** shows the output of the chi-squared test carried out using "fitdistrplus" package in R. The p-value is greater than 0.01 hence, we fail to reject the null hypothesis that the data follows a Negative binomial distribution.

In section 3, we saw that the distribution of the exceedances can be approximated by a GPD. We will test whether this theorem is justified in our dataset. We use the "bootstrap goodness-of-fit test for the GPD" [25] provided in R package "gPdtest. This test investigates the goodness-of-fit of the GPD, for cases where the distribution is heavy-tailed (shape parameter *ζ* ≥0) and non-heavy tailed (*ζ* <0).

**Figure 11.** *Density of the Exceedances.*


#### **Table 3.**

*Goodness of fit of Negative Binomial to the distribution of the number of Exceedances.*


*σ*

*σ*

*σ*

CEVD distribution is [26]:

**7.2 Goodness-of-fit tests**

*d*ð Þ *α*

*Probability-weighted Moments Estimates.*

ffiffiffiffiffiffiffiffiffiffiffiffi *n* þ *m nm* <sup>r</sup>

equations:

**Table 6.**

**Table 7.**

**291**

*Maximum Likelihood Estimates.*

*<sup>ζ</sup> <sup>α</sup> <sup>a</sup><sup>θ</sup>* 1 � *θ* � �*<sup>ζ</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.94578*

*On Modelling Extreme Damages from Natural Disasters in Kenya*

*<sup>ζ</sup> <sup>α</sup> <sup>a</sup><sup>θ</sup>* 1 � *θ* � �*<sup>ζ</sup>*

*<sup>ζ</sup> <sup>α</sup> <sup>a</sup><sup>θ</sup>* 1 � *θ* � �*<sup>ζ</sup>*

*<sup>B</sup>*ðð Þ <sup>1</sup> � *<sup>ζ</sup>* , 3ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>ζ</sup>* Þ � <sup>0</sup>*:*<sup>33</sup> ( ) <sup>¼</sup> <sup>2185841</sup>

*<sup>B</sup>*ðð Þ <sup>1</sup> � *<sup>ζ</sup>* , 4ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>ζ</sup>* Þ � <sup>0</sup>*:*<sup>25</sup> ( ) <sup>¼</sup> <sup>1997587</sup>

*B*ðð Þ 1 � *ζ* , 5ð Þ *α* þ *ζ* Þ � 0*:*2

Using the values in **Table 6** as the initial values, the MLE estimates of the NB-GP

Next, we test the goodness-of-fit of the NB-GP CEVD to the data. We create a sample of NB-GP CEVD with the parameters given in **Table 7**. Since this sample is the basis of comparison, we create a large sample of size 10,000 to closely approximate the distribution. Two-sample KS test and two-sampled AD test is then carried out to test the null hypothesis that the two samples have the same distribution. As observed in **Table 8**, the p-values in both tests are greater than 0.01. Thus, we fail to reject the null hypothesis that the two samples come from the same population at 1% level of significance. Alternatively, for the KS-test, we have:

<sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

**Parameter Estimate** *α*^ 9,000,000 ^*θ* 0.8509 *σ*^ 0.2297 ^*ζ* 0.8378 *a*^ 0.3836

**Parameter Estimate Standard Error** *α*^ 9,000,000 2.9659 ^*θ* 0.0000178 *σ*^ 1268.1920 0.0100 ^*ζ* 1.2336 0.0111 *a*^ 0.4000 0.0341

�0*:*5 ln 0ð Þ *:*<sup>1</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

10022 <sup>220000</sup> <sup>r</sup>

¼ 0*:*3239 (34)

We use "nleqslv" package in R to solve the resulting system of non-linear

¼ 1851745

(33)

( )

The likelihood function is also found to be �192*:*9873.

#### **Table 4.**

*Goodness of fit test for GPD.*

**Table 4** shows that the p-value in both cases is greater than 0.01. Hence, we conclude that the exceedances do indeed follow a GPD.
