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

#### **7.1 Parameter estimation**

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 iterative methods to estimate the parameters.

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 sample PWMs are found to be:

$$
\hat{p\_0} = 2817753, \hat{p\_1} = 2440020, \hat{p\_2} = 2185841, \hat{p\_3} = 1997587, \hat{p\_4} = 1851745 \quad \text{(31)}
$$

The resulting system of equations is then:

$$\begin{aligned} \frac{\sigma}{\zeta} \left\{ a \left( \frac{a\theta}{1-\theta} \right)^{\zeta} B((1-\zeta), (a+\zeta)) - \mathbf{1.0} \right\} &= 2817753 \\ \frac{\sigma}{\zeta} \left\{ a \left( \frac{a\theta}{1-\theta} \right)^{\zeta} B((1-\zeta), (2a+\zeta)) - \mathbf{0.5} \right\} &= 2440020 \end{aligned} \tag{32}$$


**Table 5.**

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

*On Modelling Extreme Damages from Natural Disasters in Kenya DOI: http://dx.doi.org/10.5772/intechopen.94578*

$$\frac{\sigma}{\zeta} \left\{ a \left( \frac{a\theta}{1-\theta} \right)^{\zeta} B((1-\zeta), (3a+\zeta)) - 0.33 \right\} = 2185841$$

$$\frac{\sigma}{\zeta} \left\{ a \left( \frac{a\theta}{1-\theta} \right)^{\zeta} B((1-\zeta), (4a+\zeta)) - 0.25 \right\} = 1997587 \tag{33}$$

$$\frac{\sigma}{\zeta} \left\{ a \left( \frac{a\theta}{1-\theta} \right)^{\zeta} B((1-\zeta), (5a+\zeta)) - 0.2 \right\} = 1851745$$

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

Using the values in **Table 6** as the initial values, the MLE estimates of the NB-GP CEVD distribution is [26]:

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