# Claims Frequency Distribution Models

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```					                  Claims Frequency Distribution Models

Lecture: Weeks 3-4

Lecture: Weeks 3-4 (Math 3632)     Claims Frequency Models   Spring 2010 - Valdez   1 / 20
Introduction

Introduction

Here we introduce a large class of counting distributions, which are
discrete distributions with support consisting of non-negative integers.

Generally used for modeling number of events, but in an insurance
context, the number of claims within a certain period, e.g. one year.
We call these claims frequency models.
Let N denote the number of events (or claims). Its probability mass
function (pmf), pk = Pr(N = k), for k = 0, 1, 2, . . ., gives the
probability that exactly k events (or claims) occur.

Lecture: Weeks 3-4 (Math 3632)    Claims Frequency Models   Spring 2010 - Valdez   2 / 20
Introduction    probability generating function

The probability generating function
￿∞
Recall the pgf of N : PN (z) = E(z N ) =                         k=0 pk z
k.

Just like the mgf, pgf also generates moments:
￿                  ￿￿
PN (1) = E(N ) and PN (1) = E[N (N − 1)].

More importantly, it generates probabilities:
￿ m          ￿
(m)           d
PN (z) = E          z N
= E[N (N − 1) · · · (N − m + 1)z N −m ]
dz m
￿∞
=        k(k − 1) · · · (k − m + 1)z k−m pk
k=m

Thus, we see that

(m)                                    1 (m)
PN (0) = m! pm or pm =                         P (0).
m!
Lecture: Weeks 3-4 (Math 3632)          Claims Frequency Models                          Spring 2010 - Valdez   3 / 20
Some discrete distributions

Some familiar discrete distributions

Some of the most commonly used distributions for number of claims:
Binomial (with Bernoulli as special case)
Poisson
Geometric
Negative Binomial
The (a, b, 0) class
The (a, b, 1) class

Lecture: Weeks 3-4 (Math 3632)                   Claims Frequency Models   Spring 2010 - Valdez   4 / 20
Bernoulli random variables

Bernoulli random variables

N is Bernoulli if it takes only one of two possible outcomes:
￿
1, if a claim occurs
N=                             .
0, otherwise

q is the standard symbol for the probability of a claim, i.e.
Pr(N = 1) = q.
We write N ∼ Bernoulli(q).
Mean E(N ) = q and variance Var(N ) = q(1 − q)
Probability generating function:

PN (z) = qz + (1 − q)

Lecture: Weeks 3-4 (Math 3632)                  Claims Frequency Models   Spring 2010 - Valdez   5 / 20
Binomial random variables

Binomial random variables

We write N ∼ Binomial(m, q) if N has a Binomial distribution with
pmf:
￿ ￿
m k                      m!
pk = Pr(N = k) =      q (1 − q)m−k =              q k (1 − q)m−k ,
k                   k!(m − k)!

for k = 0, . . . , m.
Binomial r.v. is also the sum of independent Bernoulli’s with
￿
N = m Nk where each Nk ∼ Bernoulli(q).
k=1

Mean E(N ) = mq and variance Var(N ) = mq(1 − q)
Probability generating function:

PN (z) = [qz + (1 − q)]m

Lecture: Weeks 3-4 (Math 3632)                 Claims Frequency Models   Spring 2010 - Valdez   6 / 20
Poisson random variables

Poisson random variables
N ∼ Poisson(λ) if pmf is

λk
pk = P (X = k) = e−λ                    , for k = 0, 1, 2, . . .
k!
Mean and variance are equal: E(N ) = Var(N ) = λ
Probability generating function of a Poisson:

PN (z) = eλ(z−1) .

Sums of independent Poissons: If N1 , . . . , Nn be n independent
Poisson variables with parameters λ1 , . . . , λn , then the sum

N = N1 + · · · + Nn

has a Poisson distribution with parameter λ = λ1 + · · · + λn .

Lecture: Weeks 3-4 (Math 3632)                Claims Frequency Models            Spring 2010 - Valdez   7 / 20
Poisson random variables   decomposition

Decomposition property of the Poisson
Suppose a certain number, N , of events will occur and
N ∼ Poisson(λ).
Suppose further that each event is either a Type 1 event with
probability p or a Type 2 event with probability 1 − p.
Let N1 and N2 be the number of Types 1 and 2 events, respectively,
so that N = N1 + N2 .
Result: N1 and N2 are independent Poisson random variables with
respective means

E(N1 ) = λp and E(N2 ) = λ(1 − p).

Proof to be provided in class.
This result can be extended to several types, say 1, 2, . . . , n, with
N = N1 + · · · + Nn .
Lecture: Weeks 3-4 (Math 3632)                Claims Frequency Models       Spring 2010 - Valdez   8 / 20
Poisson random variables   example

Example

Suppose you have a portfolio of m independent and homogeneous risks
where the total number of claims from the portfolio has a Poisson
distribution with mean parameter λ.
Suppose the number of homogeneous risks in the portfolio was changed to
m∗ .
Prove that the total number of claims in the new portfolio still has a
Poisson distribution. Identify its mean parameter.

Lecture: Weeks 3-4 (Math 3632)                Claims Frequency Models   Spring 2010 - Valdez   9 / 20
Negative binomial random variable

Negative binomial random variable
N has a Negative Binomial distribution, written N ∼ NB(β, r), if its
pmf can be expressed as
￿               ￿￿          ￿r ￿          ￿k
k+r−1              1            β
pk = Pr(N = k) =                                                               ,
k               1+β          1+β

for k = 0, 1, 2, . . . where r > 0, β > 0.
Probability generating function of a Negative Binomial:

PN (z) = [1 − β(z − 1)]−r .

Mean: E(N ) = rβ
Variance: Var(N ) = rβ(1 + β).
Clearly, since β > 0, the variance of the NB exceeds the mean.

Lecture: Weeks 3-4 (Math 3632)               Claims Frequency Models           Spring 2010 - Valdez   10 / 20
Negative binomial random variable       Geometric

Geometric random variable

The Geometric distribution is a special case of the Negative Binomial
with r = 1.
N is said to be a Geometric r.v. and written as N ∼ Geometric(p) if
its pmf is therefore expressed as
￿               ￿k
1                              β
pk = Pr(N = k) =                                                 , for k = 0, 1, 2, . . . .
1+β                            1+β

Mean is E(N ) = β and variance is Var(N ) = β(1 + β).
Its pgf is:
1
PN (z) =
1 − β(z − 1)

Lecture: Weeks 3-4 (Math 3632)               Claims Frequency Models                      Spring 2010 - Valdez   11 / 20
Negative binomial random variable   mixture of Poissons

Negative Binomial as a mixture of Poissons

Suppose that conditionally on the parameter risk parameter Λ = λ,
the random variable N is Poisson with mean λ.
To evaluate the unconditional probability of N , use the law of total
probability:
￿ ∞                              ￿ ∞ −λ k
e λ
pk =       Pr(N = k | Λ = λ)u(λ)dλ =                 u(λ)dλ,
0                                0      k!

where u(·) is the pdf of Λ.
In the case where Λ has a gamma distribution, it can be shown (to be
done in lecture) that N has a Negative Binomial distribution.
In eﬀect, the mixed Poisson, with a gamma mixing distribution, is
equivalent to a Negative Binomial.

Lecture: Weeks 3-4 (Math 3632)               Claims Frequency Models             Spring 2010 - Valdez   12 / 20
Negative binomial random variable   limiting case

Limiting case of the Negative Binomial

Prove that the Poisson distribution is a limiting case of the Negative
Binomial distribution.
Proof to be discussed in class.

Lecture: Weeks 3-4 (Math 3632)               Claims Frequency Models       Spring 2010 - Valdez   13 / 20
Negative binomial random variable   SOA question

SOA question

Actuaries have modeled auto windshield claim frequencies and have
concluded that the number of windshield claims ﬁled per year per driver
follows the Poisson distribution with parameter λ, where λ follows the
gamma distribution with mean 3 and variance 3.
Calculate the probability that a driver selected at random will ﬁle no more
than 1 windshield claim next year.

Lecture: Weeks 3-4 (Math 3632)               Claims Frequency Models      Spring 2010 - Valdez   14 / 20
Special class of distributions   the (a, b, 0) class

Special class of distributions
The (a, b, 0) class of distributions satisﬁes the recursion equations of
the general form:
pk       b
= a + , for k = 1, 2, . . . .
pk−1      k

The three distributions (including Geometric as special case of NB)
are the only distributions that belong to this class: Binomial, Poisson,
and Negative Binomial.
It can be shown that the applicable parameters a and b are:
Distribution                       Values of a and b
q                 q
Binomial(m, q)                     a = − 1−q , b = (m + 1) 1−q
Poisson(λ)                         a = 0, b = λ
β                 β
NB(β, r)                           a = 1+β , b = (r − 1) 1+β

Lecture: Weeks 3-4 (Math 3632)                Claims Frequency Models              Spring 2010 - Valdez   15 / 20
Special class of distributions   the (a, b, 0) class

Example

Suppose N is a counting distribution satisfying the recursive probabilities:
pk   4 1
= − ,
pk−1  k 3

for k = 1, 2, . . .
Identify the distribution of N .

Lecture: Weeks 3-4 (Math 3632)                Claims Frequency Models              Spring 2010 - Valdez   16 / 20
Special class of distributions   the (a, b, 0) class

SOA question

The distribution of accidents for 84 randomly selected policies is as follows:

Number of Accidents                          Number of Policies
0                                          32
1                                          26
2                                          12
3                                           7
4                                           4
5                                           2
6                                           1

Identify the frequency model that best represents these data.

Lecture: Weeks 3-4 (Math 3632)                Claims Frequency Models              Spring 2010 - Valdez   17 / 20
Truncation and modiﬁcation at zero   the (a, b, 1) class

Truncation and modiﬁcation at zero
The (a, b, 1) class of distributions satisﬁes the recursion equations of
the general form:
pk       b
= a + , for k = 2, 3, . . . .
pk−1      k

Only diﬀerence with the (a, b, 0) class is the recursion here begins at
p1 instead of p0 . The values from k = 1 to k = ∞ are the same up to
a constant of proportionality. For the class to be a distribution, the
remaining probability must be set for k = 0.
zero-truncated distributions: the case when p0 = 0
zero-modiﬁed distributions: the case when p0 > 0

The distributions in the second subclass is indeed a mixture of an
(a, b, 0) and a degenerate distribution. A zero-modiﬁed distribution
can be viewed as a zero-truncated by setting p0 = 0.

Lecture: Weeks 3-4 (Math 3632)              Claims Frequency Models             Spring 2010 - Valdez   18 / 20
Truncation and modiﬁcation at zero   the (a, b, 1) class

Their probability generating functions

We derive in class the pgf’s of zero-truncated and zero-modiﬁed
subclass of distributions.
We also discuss: “extended” truncated Negative Binomial (ETNB).
Consider also:
Example 6.8
Example 6.9

Check out Table 6.4, page 125 for summary of members of the
(a, b, 1) class.

Lecture: Weeks 3-4 (Math 3632)              Claims Frequency Models             Spring 2010 - Valdez   19 / 20
Truncation and modiﬁcation at zero   illustrative example

Illustrative example

Consider the zero-modiﬁed Geometric distribution with probabilities
1
p0 =
2
￿ ￿
1 2 k−1
pk =                 , for k = 1, 2, 3, . . .
6 3

Derive the probability generating function, the mean and the variance of
this distribution.

Lecture: Weeks 3-4 (Math 3632)              Claims Frequency Models              Spring 2010 - Valdez   20 / 20

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