Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Mixed Poisson Models for Claim Numbers 29
where is the annual expected claim number and d is the length of the observation period
(the exposure-to-risk). The probability mass function can be expressed using the generalized
binomial coefficient:
( )
a + k a a ( ) d k
PrN = k =
ak + 1 a + d a + d
( )( ) a + k − 1 a a ( ) d k
=
k= 0 1 2
k a + d a + d
Henceforth, we write N ∼ ina d to indicate that N obeys the Negative Binomial
distribution with parameters a and d. This model has been applied to retail purchasing,
absenteeism, doctor’s consultations, amongst many others.
Moments
If X ∼ am , its mean is EX = / and its variance is VX = / 2 .IfN ∼
ina d then the mean is EN = d and the variance is VN = d+d 2 /a according
to (1.29). It can be shown that / √ V = 2, in (1.31) for the Negative Binomial
distribution.
Probability Generating Function
If X ∼ am , its moment generating function is
(
Mt = 1 −
) t −
if t< (1.36)
The probability generating function of N ∼ ina d is
(
)
a
a
N z =
(1.37)
a − dz − 1
This result comes from (1.33) together with (1.36).
True and Apparent Contagion
Apparent contagion arises from the recognition that sampled individuals come from a
heterogeneous population in which individuals have a constant but different propensity to
experience accidents. A given individual may have a high (or low) propensity for accidents
but occurrence of an accident does not make it more (or less) likely that another accident will
occur. However, aggregation across heterogeneous individuals may generate a misleading
statistical finding which suggests that occurrence of an accident increases the probability of
another accident; the observed but persistent heterogeneity can be misinterpreted as due to
a strong serial dependence.
True contagion refers to dependence between the occurrences of successive events. The
occurrence of an event, such as an accident or illness, may change the probability of
subsequent occurrences of similar events. True positive contagion implies that the occurrence
of an event shortens the expected waiting time to the next occurrence of the event.
The alleged phenomenon of accident proneness can be interpreted in terms of true
contagion as suggesting that an individual who has experienced an accident is more likely