**7.2 Goodness-of-fit tests**

**Table 4** shows that the p-value in both cases is greater than 0.01. Hence, we

**Item P-value R-statistic**

<sup>0</sup> : X has a GPD with negative shape parameter 0.0961 0.9581

<sup>0</sup> :X has a GPD with positive shape parameter 0.3874 0.9720

**7. Fitting the negative binomial-generalised Pareto compound extreme**

We now fit the proposed distribution to the Kenyan data on natural disasters using MLE. We derived the log-likelihood function of the NB-GP CEVD in Section 7.1, and found that the function is not in closed form. Therefore, we will use

First, we have to select an appropriate initial value for the parameter estimates. These values represents the initial guess for the estimates. The parameters of the NB-GP CEVD are supported at *α* >0, *σ* >0, 0< *θ* <1, 0 <*a*<1 and *ζ* ∈ . We will study how the choice of the starting value affects the parameter estimates by choosing different starting points, within the support of the parameters.

From **Table 5**, we can observe how the starting values of the parameter estimates significantly affects the estimates. To get more reliable estimates, we use PWM estimates, as discussed in subsection X, to be the starting values. The fist five

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

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

**Parameter** *αθσζ* **a Starting Values 4.5 0.1 1 0.5 0.4** Estimates 24.2281 0.9228 0.9757 0.5039 0.3900 **Starting Values 0.5 0.1 1 0.5 0.4** Estimates 3.3001 0.9898 1.8597 0.7754 0.9016 **Starting Values 0.1 0.1 0.1 0.1 0.1** Estimates 0.5632 0.9781 0.2440 0.2226 0.3991

<sup>4</sup> ¼ 1851745 (31)

(32)

¼ 2817753

¼ 2440020

<sup>0</sup> ¼ 2817753, *p*^<sup>1</sup> ¼ 2440020, *p*^<sup>2</sup> ¼ 2185841, *p*^<sup>3</sup> ¼ 1997587, *p*^

( )

( )

conclude that the exceedances do indeed follow a GPD.

*Natural Hazards - Impacts, Adjustments and Resilience*

iterative methods to estimate the parameters.

The resulting system of equations is then:

*MLE estimates of the parameters using different starting values.*

*<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>*

**value distribution**

*Goodness of fit test for GPD.*

*H*�

*H*<sup>þ</sup>

**Table 4.**

**7.1 Parameter estimation**

sample PWMs are found to be:

*σ*

*σ*

*p*^

**Table 5.**

**290**

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:

$$d(a)\sqrt{\frac{n+m}{nm}} = \sqrt{-0.5\ln\left(0.1\right)}\sqrt{\frac{10022}{220000}} = 0.3239\tag{34}$$


**Table 6.**

*Probability-weighted Moments Estimates.*


**Table 7.** *Maximum Likelihood Estimates.*


**Table 8.**

*Goodness-of-fit Test results.*

The test-statistic

$$D\_{m,n} = 0.3010 < 0.3239 \tag{35}$$

so we fail to reject the null hypothesis. We conclude that the data of natural disasters in Kenya follow a NB-GP CEVD.

#### **7.3 Prediction study**

From the estimation results in **Table 7**, we can tell that the probability of damages exceeding 50,000 in any year is approximately 0.4. We now want to predict the expected maximum and minimum among those exceeding the threshold in any year with different certainties as explained in section X.

Using Eq. (X), the minimum damage that is expected to be exceeded among those exceeding the threshold with probability p, is predicted using:

$$\hat{\alpha}\_p = 50000 + \frac{\hat{\sigma}}{\hat{\zeta}} \left\{ \left[ \frac{\hat{a}\hat{\theta}}{(1-\hat{\theta})\left((1-p)^{-1/\hat{a}} - 1\right)} \right]^\zeta - 1 \right\} \tag{36}$$

The corresponding minimum equation, *xp*,among those exceeding 50000 that is expected not to be exceeded at any given year under different probabilities can be predicted using:

$$\hat{\mathbf{x}}\_p = \mathbf{50000} + \left\{ \frac{\hat{\sigma}}{\hat{\zeta}} \left[ \frac{\hat{a}\hat{\theta}}{(1-\hat{\theta})(p^{-1/\hat{\alpha}}-1)} \right]^\zeta - 1 \right\} \tag{37}$$

The Relative error is highest at 0,9 level of certainty and lowest at 0.999 certainty level. So, accuracy of the predictions increases as the level of certainty increases. Generally, the distribution performs well at predicting the minimum values of those

**Probability 0.9 0.95 0.99 0.995 0.999** Absolute Error for the Minimum 50,293.000 33,241.663 14,917.407 10,753.149 4,594.092 Relative Error for the Minimum 0.833 0.551 0.247 0.178 0.076 Absolute Error for the Maximum 20,543,049 16,642,274 26,020,717 92,724,979 818,675,727 Relative Error for the Maximum 0.880 0.713 1.115 3.974 35.089

**Probability 0.9 0.95 0.99 0.995 0.999**

*On Modelling Extreme Damages from Natural Disasters in Kenya*

**Statistic Value** Minimum 60,340 Median 692,142 Mean 2,817,753 Maximum 23,331,469

2,788,420.00 6,689,195.00 49,35,2186.00 116,056,448.00 842,007,196.00

110,633.00 93,581.66 75,257.41 71,093.15 64,934.09

0.9 and 0.95 probabilities, and higher at *p*≥ 0*:*99. So, the proposed distribution underestimates the maximum values at probability levels less than 0.95, and overestimates them at probability levels greater than 0.95. In addition, the relative maximum errors are smaller when the predicted values are lower than the actual ones, as compared to when they are greater. So, the distribution overestimates the maximum values by a greater margin than it tends to underestimate. IN general, the NB-GP CEVD performs poorly at predicting the maximum values of the damages

In overall, the absolute errors are smaller for the minimum values than the maximum values. This implies that the NB-GPD CEVD performs better at

predicting the minimum values among those exceeding the threshold, than it does

The main goal for this research is to model the extreme damages in Kenya resulting from natural disasters by considering both occurrence and the magnitude

For the maximum values, the predicted values are lower than the actual values at

exceeding the threshold.

*Prediction Accuracy for Extreme Damages.*

Predicted Maximum

Predicted Minimum

*Prediction Results for Extreme Damages in Kenya.*

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

Value

Value

**Table 9.**

**Table 10.**

**Table 11.**

*Statistics of the Exceedances.*

exceeding the threshold.

for the maximum values.

**8. Conclusions**

**293**

Using the parameter estimates in **Table 7**, the prediction results for *p* ¼ 0*:*9, 0*:*95, 0*:*99, 0*:*995, 0*:*999 are given in **Table 9**.

The predicted maximum damage increases as the probabilities increase while the predicted minimum damage decreases as the probabilities increase. In other words, the range becomes larger at higher certainty levels. We can determine the accuracy of the prediction by comparing the prediction results to the statistics of the exceedances (**Table 10**). This can be done using two measures:

1.Absolute Error = Predicted Value - Actual Value

2.Relative Error = AbsoluteError ActualValue

From **Table 11**, all the error measurements for the minimum values are positive, suggesting that the predicted values are higher than the actual value. This implies that the NB-GP CEVD tends to overestimate the minimum damage among the exceedances in any given year. We can also observe how the relative minimum error for the the minimum damage decreases as the probability increases.
