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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