Estimation, Evaluation, and Selection of Actuarial Models

20 CHAPTER 2. MODEL ESTIMATION
In each case there is a parameter that relates to the spread of the kernel. In the first two cases
it is the value of b>0, which is called the bandwidth. In the gamma case, the value of α controls
the spread with a larger value indicating a smaller spread. There are other kernels that cover the
range from zero to infinity.
Exercise 12 Provide the formula for the Pareto kernel.
Example 2.21 Determine the kernel density estimate for Example 2.8 using each of the three
kernels.
The empirical distribution places probability 1/8 at 1.0, 1/8 at 1.3, 2/8 at 1.5, 3/8 at 2.1, and
1/8 at 2.8. For a uniform kernel with a bandwidth of 0.1 we do not get much separation. The data
point at 1.0 is replaced by a horizontal density function running from 0.9 to 1.1 with a height of
1 1
8 2(0.1)
=0.625. On the other hand, with a bandwidth of 1.0 that same data point is replaced by a
horizontal density function running from 0.0 to 2.0 with a height of 1 8 1
2(1)
=0.0625. The following
two figures provide plots of the density functions.
f(x )
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 1 2 3 4 5
x
f(x )
0.6
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 4 5
x
Uniform kernel density with bandwidth 0.1 Uniform kernel density with bandwidth 1.0
It should be clear that the larger bandwidth provides more smoothing. In the limit, as the bandwidth
approaches zero, the kernel density estimate matches the empirical estimate. Note that if the
bandwidth is too large, probability will be assigned to negative values, which may be an undesirable
result. Methods exist for dealing with that issue, but they will not be presented in this Note.
For the triangular kernel, each point is replaced by a triangle. Pictures for the same two
bandwidths used above appear below.
f(x )
4
3.5
3
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5
x
f(x )
0.6
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 4 5
x
Triangular kernel density with bandwidth 0.1 Triangular kernel density with bandwidth 1.0
Once again, the larger bandwidth provides more smoothing. The gamma kernel simply provides a
mixture of gamma distributions (see the Part 3 Note) where each data point provides the mean and