Example: Insurance data

We consider the data from casualty insurances, given in Example 2.6.1 with theX

value representing Loss and theY value ALAE (allocated loss adjustment expenses). We are interested in the event that the sum of the two values for the next observation is greater than a certain value t, so, Tn+1 = Xn+1+Yn+1 > t. In this example we

used only the bandwidth selections and the types of bandwidths discussed in Section 3.3.1, which did not lead to problem with thehij values, i.e. normal reference rule-

of-thumb and LSCV bandwidth selections, with fixed and adaptive-nn bandwidths. The results are presented in Table 3.7.

Bandwidth selection Type of bandwidth bX bY

Normal Reference rule-of-thumb fixed 0.1647 0.1647

Least Square Cross Validation fixed 0.1673 0.2577

average adaptive-nn 0.5329 0.9325

Table 3.7: Bandwidth selections and type of bandwidths

The results in Table 3.7 show that the bandwidths of the Loss and ALAE vari- ables are 0.1647 when using the normal reference rule-of-thumb. The value is the same for both variables because its a fixed bandwidth type. For the LSCV method, the fixed bandwidth type gives bandwidth for Loss 0.1673 and for ALAE 0.2577. The bandwidths for these two random quantities are different because the LSCV method chooses the bandwidth based on minimizing the integrated squared error as discussed in Section 3.2. For Loss and ALAE variables, the algorithm produced 4-th and 7-th adaptive-nn, respectively. The corresponding bandwidth values are

b = 0.5329 and b = 0.9325, respectively, using the formula given in Section 3.2 i.e.

bz = kzσzn(−1/4). Figure 3.5 shows lower and upper probabilities according to our

method for the event Tn+1 > t corresponding to these bandwidth selections and

3.3. Combining NPI with kernel-based copula 59 depending on applications or events of interest. From this figure we cannot see much differences, it seem all bandwidth selections and types of bandwidth used give an identical figure of lower and upper probabilities for the event Tn+1 > t. However,

the bandwidths obtained in Table 3.7 shows that the LSCV method gives higher bandwidth for ALAE compared to normal reference rule-of thumb which shows the LSCV method is over-smoothing the probabilities hij. As the probabilities hij are

the most important part in this study, we show the 3D-plot of this data set for all bandwidth selections and types of bandwidths in Figure 3.6. From this figure, the probabilities hij are quite higher at left-front corner and right-back corner for

each subfigure (i.e. Figures 3.5(a), 3.5(b) and 3.5(c)), and the probabilities hij are

scattered at most of the cells for all bandwidth selections and the types of band- widths. However, the probabilitieshij are different in small amount among the cells.

These features are the reason that the lower and upper probabilities for the event

Tn+1 > t is quite identical in Figure 3.5. Another noticeable feature when the pro-

posed method applied for this data set, the probabilities hij are more scattered in

(a) Normal reference rule-of-thumb; fixed bandwidth

(b) LSCV; fixed bandwidth

(c) LSCV; Adaptive-nn bandwidthkx= 4 and

ky= 7

3.3. Combining NPI with kernel-based copula 61

(a) Normal reference rule-of-thumb; fixed bandwidth

(b) LSCV; fixed bandwidth

(c) LSCV; Adaptive-nn bandwidth kx = 4

andky= 7

Figure 3.6: 3D-plot of probabilitieshij for bandwidth selections and types of band-

widths

Due to quite identical lower and upper probabilities for the event Tn+1 > t in

Figure 3.5, we use different values of kz for the adaptive-nn bandwidth. kz is the

kth nearest neighbours of the observations and it is related to the distance of any point to its nearest observations as discussed in Section 3.2. We only use this type of bandwidth because this method allows us to determine the nearest neighbour to be used, while the other types of bandwidth do not offer this possibility. Another reason why we investigate this method further for different values ofkz is, as mentioned in

Section 3.2, estimation is influenced by the values of kz, so, the values of kz might

also affect prediction. Figure 3.7 shows the lower and upper probabilities for the event Tn+1 > t for our method, for different values ofkz. From this figure, we can

see that as kz increases, the lower and upper survival functions become smoother.

This is due to the adaptive-nn bandwidth used in our method and the specific event of interest. For example, when we consider kz = 2 in adaptive-nn bandwidth, this

type of bandwidth uses two nearest points from the point that need to be estimated, and this gives a few peaks. If we consider kz = 5, the adaptive-nn bandwidth uses

the five nearest points from the point that need to be estimated, and this gives fewer peaks. In other words, the 5-th nearest neighbour uses a broader distance for estimating the points. In addition, as we consider the sum event of the bivariate random quantities, the possibility probabilitieshij to be included or not is depending

on how the probabilities hij are scattered. We show the 3D-plots of probabilities

hij for kz = 2 and kz = 5 in Figure 3.8. Figure 3.8 shows that the probabilities

hij are scattered differently between kz = 2 and kz = 5, whereby the probabilities

hij are higher at kz = 2 compared to kz = 5. So, these 3D-plots suggest that

the kz for adaptive-nn bandwidth does affect the prediction. This is due to the

nearest point used and the way of adaptive-nn bandwidth work. As the value ofkz

increases, the adaptive-nn method over-smooth the probabilitieshij. Consequently,

the three conditions for the probabilitieshij discussed in Section 2.3 are not satisfied.

In the following section, we will investigate the bandwidth selection related to the predictive performance of our method.

3.3. Combining NPI with kernel-based copula 63

(a)kx=ky = 1 (b) kx=ky = 2

(c) kx=ky= 3 (d) kx=ky = 4

(e) kx=ky= 5 (f) kx=ky= 6

Figure 3.7: Lower and upper probabilities for the event Tn+1 > t, adaptive-nn

(a)kz= 2 (b) kz= 5

Figure 3.8: 3D-plot of probabilitieshij forkx =ky = 2 andkx =ky = 5 adaptive-nn

bandwidth.

3.4

Predictive performance

We conducted a simulation study to obtain an indication of the predictive perfor- mance of this method. We used a similar method as discussed in Section 2.5 to indicate the predictive performance of our method, but now using a nonparametric copula. The results are based on N = 10,000 bivariate simulated samples, each

of size n + 1, using the Frank, Normal, Clayton and Gumbel copulas. For each

simulated sample, the first n pairs are used as data for our predictive method, the additional pair is considered as a future observation and is used to test the predic- tive performance of this method. Equations (2.9) and (2.10) in Section 2.5 are used to indicate the performance of the proposed method. In other words, the proposed method performs well if the two inequalities in equations (2.9) and (2.10) hold.

Based on previous example in Section 3.3.2, we conducted two types of simulation studies. First, in Section 3.4.1 we use auto-driven bandwidth selection where we let algorithm namely npudistbw in the R package np [49] choose the bandwidth. Secondly, in Section 3.4.2 we use manual bandwidth selection where we choose the

3.4. Predictive performance 65 search algorithm uses direction set (Powell [78]) methods in multidimensions [49] to optimize the bandwidth. In thenppackage, the optimizer used is Powell’s conjugate direction method, which requires the setting of initial values and search directions for bandwidths, and when restarting, random values for successive invocations [49].