Question

Two insureds own delivery vans. Insured A had two vans in year 1 and one claim, two vans in year 2 and one claim, and one van in year 3 with no claims. Insured B has no vans in year 1, three vans in year 2 and two claims, and two vans in year 3 and three claims. The number of claims for insured each year has a Poisson distribution. Use semiparametric empirical Bayes estimation to obtain the estimated number of claims for each insured in year 4.

Answer 1

Answer:

For A: 2.208

For B: 2.9024

Step-by-step explanation:

To solve we will do the following:

For A:

no insured(x)      claims(y)         xy     Var=total/(y-1)

        2                       1                  2                   0.08

        2                       1                  2                   0.08

        1                        0                 0                   0.64

Weighted average                      0.8                 0.16  

        a                                    var - X            -0.64  

        k                                   mean/a            -1.25  

Z (credibility)                              1/(1+k)        0.444

no of claims for next year (y=4)

= 0.44*4 + 0.56*0.8

= 2.208

For B:

no insured(x)      claims(y)         xy     Var=total/(y-1)

        0                       0                 0                     0

        3                       6                 12                    12

        2                        3                 6                     2

Weighted average                      2.4                  3.5

        a                                    var - X              1.1  

        k                                   mean/a              2.18  

Z (credibility)                              1/(1+k)        0.314

no of claims for next year (y=4)

0.314*4 + 0.686*2.4

= 2.9024