Operational Risk Modeling Analytics phần 4 ppt

122 MODELS FOR THE NUMBER OF LOSSES: COUNTlNG DISTRIBUTIONS It is also interesting that the special extreme case with -1 < r < 0 and p -+ 00 is a proper distribution, sometimes called th

THE (a, b, 1) CLASS 121

If the original values were ail available, then the zero-truncated probabilities could have all been obtained by multiplying the original values by 1/(1 - 0.362887) = 1.569580

For the zero-modified random variable, pf = 0.6 arbitrarily From (5.4),

p r = (1 - 0.6)(0.302406)/(1 - 0.362887) = 0.189860 Then

p? = 0.189860 (5 + $+) = 0.110752,

p y = 0.110752 (5 + $4) = 0.055376

In this case, each original negative binomial probability has been multiplied

by (1 - 0.6)/(1 - 0.362887) = 0.627832 Also note that, for j 2 1, p y =

Although we have only discussed the zero-modified distributions of the (a, b, 0) class, the (a, b, 1) class admits additional distributions The (a, b) parameter space can be expanded to admit an extension of the negative bi- nomial distribution to include cases where -1 < T < 0 For the (a, b, 0) class,

T > 0 is required By adding the additional region to the sample space, the

“extended” truncated negative binomial (ETNB) distribution has parameter restrictions ,B > 0, T > -1, T # 0

To show that the recursive equation

p k = p k - l ( U S - 3 , k = 2 , 3 , , (5.8) with po = 0 defines a proper distribution, it is sufficient to show that for any value of pl , the successive values of pk obtained recursively are each positive and that C&pk < co For the ETNB, this must be done for the parameter

122 MODELS FOR THE NUMBER OF LOSSES: COUNTlNG DISTRIBUTIONS

It is also interesting that the special extreme case with -1 < r < 0 and

p -+ 00 is a proper distribution, sometimes called the Sibuya distribution

It has pgf P ( z ) = 1 - (1 - z ) - ~ , and no moments exist (see Exercise 5.8) Distributions with no moments are not particularly interesting for modeling loss numbers (unless the right tail is subsequently modified) because an infinite number of losses are expected If this is the case, the risk manager should be fired!

Example 5.7 Determine the probabilities for an ETNB distribution with r = -0.5 and /3 = 1 Do this both for the truncated version and for the modified version with p f = 0.6 set arbitrarily

We have a = 1/(1 + 1) = 0.5 and b = (-0.5 - 1)(1)/(1 + 1) = -0.75 We also have p r = -0.5(1)/[(1 + l)0.5 - (1 + l)] = 0.853553 Subsequent values are

p; = ( 0.5 - - O,,) (0.853553) = 0.106694,

p; = ( 0.5 - - O,,) (0.106694) = 0.026674

For the modified probabilities, the truncated probabilities need to be multi- plied by 0.4 to produce p y = 0.341421, p y = 0.042678, and p y = 0.010670 Note: A reasonable question is to ask if there is a “natural” member of the ETNB distribution, that is, one for which the recursion would begin with pl

rather than pa For that to be the case, the natural value of po would have

to satisfy pl = (0.5 - 0.75/l)p0 = -0.25~0 This would force one of the two probabilities to be negative and so there is no acceptable solution It is easy

0

to show that this occurs for any r < 0

There are no other members of the (a, b, 1) class beyond those discussed above A summary is given in Table 5.4

5.7 COMPOUND FREQUENCY MODELS

A larger class of distributions can be created by the processes of compounding any two discrete distributions The term compounding reflects the idea that

the pgf of the new distribution P(z) is written as

P ( z ) = PrV LPM ( z ) ] 7 (5.11) where PN(z) and PM ( z ) are called the primary and secondary distributions, respectively

The compound distributions arise naturally as follows Let N be a count- ing random variable with pgf PN(z) Let M I , M2, be identically and

COMPOUND FREQUENCY MODELS 123

Table 5.4 Members of the (a, b, 1) class

bExcluding T = 0, which is the logarithmic distribution

independently distributed counting random variables with pgf PM ( 2 ) As- suming that the Mjs do not depend on N , the pgf of the random sum S =

M I + M2 + + MN (where N = 0 implies that S = 0) is Ps ( Z ) = PN [PM ( 2 ) )

This is shown as

124 MODELS FOR THE NUMBER OF LOSSES: COUNTlNG DlSTRlBUTlONS

the number of losses (errors, injuries, failures, etc.) from the events, then

S represents the total number of losses for all such events This kind of in- terpretation is not necessary to justify the use of a compound distribution

If a compound distribution fits data well, that may be enough justification itself Also, there are other motivations for these distributions, as presented

in Section 5.9

Example 5.8 Demonstrate that any zero-modified distribution is a compound distribution

Consider a primary Bernoulli distribution It has pgf PN(z) = 1 - q + 42

Then consider an arbitrary secondary distribution with pgf PM(z) Then, from formula (5.11) we obtain

PS(z) = PN[PM(z)] = 1 - q + q p M ( z ) From formula (5.3), it is clear that this is the pgf of a ZM distribution with

This distribution is called the Poisson-Poisson or Neyrnan Type A distri-

bution Let PN(z) = e and Phf(z) = eA2('-') Then

COMPOUND FREQUENCY MODELS 125

When X2 is a lot larger than X I (for example, XI = 0.1 and Xz = 10) the

17 resulting distribution will have two local modes

Example 5.10 Demonstrate that the Poisson-logarithmic distribution is a negative binomia1,as compound Poisson-logarithmic distribution

The negative binomial distribution has pgf

where r = A/ ln(l+P) This shows that the negative binomial distribution can

be written as a compound Poisson distribution with a logarithmic secondary

Now, PM(T) can always be written as

h f ( z ) = fo + (1 - fo)Pif(.) (5.12) where P&(z) is the pgf of the conditional distribution over the positive range (in other words, the zero-truncated version)

Theorem 5.11 Suppose the p g f P N ( z ; 0) satisfies

PN(z; 0) = B[O(Z - l)]

for some parameter 0 and some function B ( z ) that is independent of0 That

is, the parameter 0 and the argument z only appear in the pgf as O(z - 1)

126 MODELS FOR THE NUMBER OF LOSSES: COUNTING DISTRIBUTIONS

There may be other parameters as well, and they may appear anywhere in the

pgf Then Ps(z) = P l v [ P ~ ( z ) ; 61 can be rewritten as

5.8 RECURSIVE CALCULATION OF COMPOUND PROBABILITIES

The probability of exactly k losses can be written as

where fin, k = O , l , , is the n-fold convolution of the function f k , k =

0,1, ., that is, the probability that the sum of n random variables which are each independent and identically distributed (iid) with probability function

f k will take on value k

oi)

n = O

RECURSIVE CALCULATION OF COMPOUND PROBABILITIES 127

When P ~ y ( z ) is chosen to be a member of the (a, b, 0) class,

128 MODELS FOR THE NUMBER OF LOSSES: COUNTING DlSTRlBUTlONS

Therefore,

Rearrangement yields the recursive formula (5.16) 0

This recursion (5.16) has become known as the Panjer recursion after its introduction as a computational tool for aggregate losses by Panjer [88] Its use here is numerically equivalent to its use for aggregate losses in Chapter 6

In order to use the recursive formula (5.16), the starting value go is required and is given in Theorem 5.15

Theorem 5.13 If the primary distribution is a member of the (a, b, 1) class, the recursive formula is

[pl - (a + b)PO]fk + c:=, (a + b j / k ) f j g k - j

g k = , k = 1 , 2 , 3 , (5.17) Proof: It is similar to the proof of Theorem 5.12 and is left to the reader 0

Example 5.14 Develop the Panjer recursive formula for the case where the primary distribution is Poisson

1 - afo

In this case a = 0 and b = A, yielding the recursive form

The starting value is, from (5.11),

go = P r ( S = 0 ) = P(O)

= e v j P M ( 0 ) l = P N ( f 0 )

-

- , - W - f o ) (5.18) Distributions of this type are called compound Poisson distributions’ When the secondary distribution is specified, the compound distribution is called

0

Poisson-X, where X is the name of the secondary distribution

The method used to obtain go applies to any compound distribution Theorem 5.15 For any compound distribution, go = PN( f o ) , where PN(z) is the pgf of the primary distribution and fo is the probability that the secondary distribution takes on the value zero

‘In some textbooks, the term conipound distribution, as in “compound Poisson,” refers to what are called in this book “mixed distributions.”

RECURSIVE CALCULATION O f COMPOUND PROBABILITIES 129

Note that the secondary distribution is not required to be in any special form However, to keep the number of distributions manageable, secondary distributions will be selected from the (a, b, 0) or the (a, b, 1) class

Example 5.16 Calculate the probabilities for the Poisson-ETNB distribution where X = 3 for the Poisson distribution and the E T N B distribution has

r = -0.5 and f l = 1

From Example 5.7 the secondary probabilities are fo = 0, f1 = 0.853553,

f2 = 0.106694, and f3 = 0.026674 From equation (5.18), go = exp[-3(1 - O)] = 0.049787 For the Poisson primary distribution, a = 0 and b = 3 The recursive formula (5.16) becomes

X can be any distribution

Example 5.17 Determine the probabilities for a Poisson-zero-modified E T N B distribution where the parameters are X = 7.5, p f = 0.6, r = -0.5, and /3 = 1 From Example 5.7 the secondary probabilities are fo = 0.6, f l = 0.341421,

f 2 = 0.042678, and f3 = 0.010670 From equation (5.18), go = exp[-7.5(1 - 0.6)] = 0.049787 For the Poisson primary distribution, a = 0 and b = 7.5 The recursive formula (5.16) becomes

130 MODELS FOR THE NUMBER OF LOSSES: COUNTING DISTRIBUTIONS

5.9 A N INVENTORY OF DISCRETE DISTRIBUTIONS

In the previous sections of this chapter, we have introduced the simple (a, b, 0) class, generalized to the (a, b, 1) class, and then used compounding to create

a larger class of distributions In this section, we summarize the distributions introduced in those sections

There are relationships among the various distributions similar to those of Section 4.3.2 The specific relationships are given in Table 5.5

It is clear from earlier developments that members of the (a, b,O) class are special cases of members of the (a, b, 1) class and that zero-truncated distributions are special cases of zero-modified distributions The limiting cases are best discovered through the probability generating function, as was done on page 113, where the Poisson distribution is shown to be a limiting case of the negative binomial distribution

We have not listed compound distributions where the primary distribution

is one of the two parameter models such as the negative binomial or Poisson- inverse Gaussian This was done because these distributions are often them- selves compound Poisson distributions and, as such, are generalizations of distributions already presented This collection forms a particularly rich set

of distributions in terms of shape However, many other distributions are also possible Many others are discussed in Johnson, Kotz, and Kemp [65], Douglas [24], and Panjer and Willmot [93]

The (a, b, 0) class

AN INVENTORY OF DISCRETE DISTRIBUTIONS 131

Table 5.5 Relationships among discrete distributions

Distribution Is a special case of Is a limiting case of Poisson

ZM geometric

Z T negative binomial

ZM negative binomial

Negative binomial Poisson-binomial Poisson-inv Gaussian Polya-Aepplia

Neyman-Ab

ZT negative binomial

ZM negative binomial Geometric-Poisson

ZT negative binomial

ZM negative binomial

ZM binomial

ZM negative binomial, Poisson-ETNB

Poisson-ETNB Poisson-ETNB

Poisson-ETNB aAlso called Poisson-geometric

bAlso called Poisson-Poisson

132 MODELS FOR THE NUMBER OF LOSSES: COUNTING DISTRIBUTIONS

To distinguish this class from the (a, b, 0) class, the probabilities are denoted

P r ( N = k ) = p p or Pr(N = k ) = p z depending on which subclass is being represented For this class, p f is arbitrary (that is, it is a parameter) and then p v or pT is a specified function of the parameters a and b Subsequent probabilities are obtained recursively as in the (a, b, 0) class: p p = (u +

b/Ic)pE1, k = 2 , 3 , ., with the same recursion for p; There are two sub- classes of this class When discussing their members, we often refer to the

“corresponding” member of the (a, b, 0) class This refers to the member of that class with the same values for a and b The notation Pl, will continue to

be used for probabilities for the corresponding (a, b, 0) distribution

The (a, b, 1) class

5.9.3 The zero-truncated subclass

The members of this class have p: = 0 and therefore it need not be estimated These distributions should only be used when a value of zero is impossible The first factorial moment is p(1) = (a + b ) / [ ( l - a ) ( l - P O ) ] , where po is the value for the corresponding member of the (a, b, 0) class For the logarithmic distribution (which has no corresponding member), p(1) = p/ In(l+P) Higher factorial moments are obtained recursively with the same formula as with the (a, b, 0) class The variance is ( a + b ) [ l - (u + b + l ) p o ] / [ ( l - a ) ( l - po)12.For

those members of the subclass that have corresponding (a, b, 0) distributions, P; = P d ( 1 -Po)

AN INVENTORY OF DISCRETE DISTRIBUTIONS 133

5.9.3.1 Zero-truncated Poisson

5.9.3.2 Zero-truncated geometric

PT =

P: = E[N] =

134 MODELS FOR THE NUMBER OF LOSSES: COUNTING DISTRIBUTIONS

5.9.4 The zero-modified subclass

A zero-modified distribution is created by starting with a truncated distri- bution and then placing an arbitrary amount of probability at zero This

probability, p f , is a parameter The remaining probabilities are adjusted accordingly Values of p f can be determined from the corresponding zero- truncated distribution as p f = (1 - p f ) p z or from the corresponding ( a , b, 0) distribution as p f = (1 - pf)pk/(l - PO) The same recursion used for the

zero-truncated subclass applies

The mean is 1 - p f times the mean for the corresponding zero-truncated distribution The variance is 1 - pf times the zero-truncated variance plus py(1-pf) times the square of the zero-truncated mean The probability gen- erating function is PM(z) = p? +(1 -pf)P(z), where P(z) is the probability generating function for the corresponding zero-truncated distribution

AN INVENTORY OF DISCRETE DlSTRlBUTlONS 135

5.9.5 The compound class

Members of this class are obtained by compounding one distribution with another That is, let N be a discrete distribution, called the primary distri- bution and let M I , M 2 , be identically and independently distributed with another discrete distribution, called the secondary distribution The com-

pound distribution is S = M I +- .+ M N The probabilities for the compound distributions are found from the Panjer recursion

k

for k = 1,2, ., where a and b are the usual values for the primary distribution [which must be a member of the (a, b, 0) class] and f j is the probability from the secondary distribution The only two primary distributions listed here are Poisson (for which po = exp[-X(l - fo)]) and geometric [for which po = l/[l+P-pfo]] The probability generating function is P ( z ) = f " [ P ~ ( z ) ] In the following list the primary distribution is always named first For the first, second, and fourth distributions, the secondary distribution is the (a, b, 0) class member with that name

5.9.5.1 Poisson-binomial

This distribution has a Poisson primary distribution and a binomial secondary or, equivalently, a Poisson primary and a zero-truncated sec- ondary distribution

5.9.5.2 Poisson-Poisson

The parameter A1 is for the primary Poisson distribution, and X2 is for the secondary Poisson distribution This distribution is also called the

Neyman Type A

5.9.5.3 Geometric-extended truncated negative binomial

The parameter ,& is for the primary geometric distribution The last two parameters are for the secondary distribution, noting that for T = 0 the secondary distribution is logarithmic The truncated version is used

so that the extension of r is available

136 MODELS FOR THE NUMBER OF LOSSES: COUNTING DISTRIBUTIONS

When r = 0 the secondary distribution is logarithmic, resulting in the negative binomial distribution This distribution is also called the gen- eralized Poisson-Pascal

5.10 A HIERARCHY OF DISCRETE DISTRIBUTIONS

The following table indicates which distributions are special or limiting cases

of others For the special cases, one parameter is set equal t o a constant to create the special case For the limiting cases, two parameters go to infinity

or zero in some special way

Distribution Is a special case of Is a limiting case of Poisson

Negative binomial, Poisson-binomial, Poisson-inv Gaussian, Polya-Aeppli,

Ne y man- A

ZT negative binomial

ZM negative binomial Geometric-Poisson

ZT negative binomial

ZM negative binomial Poisson-ETNB Poisson-ETNB

FURTHER PROPERTIES OF THE COMPOUND POISSON CLASS 137

5.11 FURTHER PROPERTIES OF THE COMPOUND POISSON CLASS

Of central importance within the class of compound frequency models is the class of compound Poisson frequency distributions Physical motivation for this model arises from the fact that the Poisson distribution is often a good model to describe the number of loss-causing accidents, and the number of losses from an accident is often itself a random variable In addition, there are numerous convenient mathematical properties enjoyed by the compound Poisson class In particular, those involving recursive evaluation of the prob- abilities were also discussed in Section 5.9.5 In addition, there is a close connection between the compound Poisson distributions and the mixed Pois- son frequency distributions which is discussed in more detail in Section 5.13 Here we consider some other properties of these distributions The compound Poisson pgf may be expressed as

where Q(z) is the pgf of the secondary distribution

Example 5.18 Obtain the pgf for the Poisson-ETNB distribution and show that it looks like the pgf of a Poisson-negative binomial distribution

The ETNB distribution has pgf

[ I - p(z - 1)]+ - (1 +

Q(z) = 1 - ( 1 + /3)+

for P > 0, r > -1, and r # 0 Then the Poisson-ETNB distribution has as

the logarithm of its pgf

to estimation and analysis of the parameters

138 MODELS FOR THE NUMBER OF LOSSES: COUNTING DISTRIBUTIONS

We can compare the skewness (third moment) of these distributions to develop an appreciation of the amount by which the skewness, and hence the tails of these distributions, can vary even when the mean and variance are fixed From equation (5.19) (see Exercise 5.14) and Definition 2.18, the mean and second and third central moments of the compound Poisson distribution are

Negative binomial: p3 = 3a2 - 2p + 2 - 'I2

Note that for fixed mean and variance the third moment only changes through the coefficient in the last term for each of the five distributions For the Poisson distribution, p3 = X = 3a2 - 2p, and so the third term for each expression for p 3 represents the change from the Poisson distribution For the Poisson-binomial distribution, if m = 1, the distribution is Poisson because

it is equivalent to a Poisson-zero-truncated binomial as truncation at zero

FURTHER PROPERTIES OF THE COMPOUND POISSON CLASS 139

leaves only probability at 1 Another view is that from the formula for the third moment (5.21), we have

Example 5.19 The data in Table 5.6 are taken from Hossack et al [55] and give the distribution of the number of losses on accidents involving automobiles

in Australia Determine an appropriate frequency model based on the skewness results of this section

The mean, variance, and third central moment are 0.1254614, 0.1299599,

and 0.1401737, respectively For these numbers,

p3 - 3a2 + 2p

(a2 - P I 2 / P = 7.543865

From among the Poisson-binomial, negative binomial, Polya-Aeppli, Neyman Type A, and Poisson-ETNB distributions, only the latter is appropriate For this distribution, an estimate of r can be obtained from

140 MODELS FOR THE NUMBER OF LOSSES: COUNTING DISTRIBUTIONS

Table 5.6 Hossack et al data

{qn; n = 0,1,2, }, where qn = [Aiqn(l) + Azqn(2) + + A k q n ( k ) ] / A

Proof: Let Q i ( z ) = Cr=oqn(i)zn for i = 1 , 2 , , k Then Si has pgf

P , , ( z ) = E(zsS) = exp{Ai[Qi(z) - 11) Because the Sis are independent,

One main advantage of this result is computational If we are interested

in the sum of independent compound Poisson random variables, then we do not need to compute the distribution of each compound Poisson random vari- able separately (i.e., recursively using Example 5.14) because Theorem 5.20 implies that a single application of the compound Poisson recursive formula

in Example 5.14 will suffice The following example illustrates this idea

Example 5.21 Suppose that k = 2 and 5’1 has a compound Poisson distri- bution with A1 = 2 and secondary distribution qI(1) = 0.2,q2(1) = 0.7, and q3(1) = 0.1 Also, 5’2 (independent of 5’1) has a compound Poisson distrib- ution with A2 = 3 and secondary distribution 42(2) = 0.25,q3(2) = 0.6, and q4(2) = 0.15 Determine the distribution of S = S1 + Sz

FURTHER PROP€RJ/ES OF THE COMPOUND POISSON CLASS 141

We have X = X I + X2 = 2 + 3 = 5 Then

41 = 0.4(0.2) + 0.6(0) = 0.08, q2 = 0.4(0.7) + O.s(O.25) = 0.43,

43 = 0.4(0.1) + O.S(O.6) = 0.40,

q4 = 0.4(0) + 0.6(0.15) = 0.09

Thus, S has a compound Poisson distribution with Poisson parameter X = 5 and secondary distribution q1 = 0.08,qz = 0 4 3 , ~ = 0.40, and 4 4 = 0.09 Numerical values of the distribution of S may be obtained using the recursive formula

beginning with P r ( S = 0) = eP5

In various situations the convolution of negative binomial distributions is

of interest Example 5.22 indicates how this distribution may be evaluated

Example 5.22 (Convolutions of negative binomial distributions) Suppose that Ni has a negative binomial distribution with parameters ri and pi for

i = 1,2, , k and that N I , N2, , Nk are independent Determine the dis- tribution of N = N1 + Nz + + Nk,

The pgf of Ni is Pjv,(z) = 11 - p i ( z - 1)ILT1 and that of N is PN(z) =

n i = , P ~ , ( z ) = nF=,[l - pi(z - l)]-Tz If pi = p for i = 1 , 2 , ,k, then

P N ( z ) = [l - P(.z - 1)]-(T1+T2+"'+Tk), and N has a negative binomial distrib- ution with parameters r = r1 + r2 + + rk and p

If not all the pis are identical, however, we may proceed as follows From Example 5.10,

k

pjv,(z) = [1 - pi(z - 1)]-~1 = e ~ ~ [ Q ~ ( z ) - ' I where X i = ri ln(1 + pz) and

with

But Theorem 5.20 implies that N = Nl + N2 + + Nk has a compound Poisson distribution with Poisson parameter

142 MODELS FOR THE NUMBER OF LOSSES: COUNTING DISTRIBUTIONS

and secondary distribution

The distribution of N may be computed recursively using the formula

5.12 MIXED FREQUENCY MODELS

Many compound distributions can arise in a way that is very different from compounding In this section, we examine mixture distributions by treating one or more parameters as being “random” in some sense This section ex- pands on the ideas discussed in Section 5.3 in connection with the gamma mixture of the Poisson distribution being negative binomial

We assume that the parameter is distributed over the population under consideration (the collective) and that the sampling scheme that generates

our data has two stages First, a value of the parameter is selected Then, given that parameter value, an observation is generated using that parameter value

Let P(zj8) denote the pgf of the number of events (e.g., losses) if the risk parameter is known to be 6 The parameter, 8, might be the Poisson mean, for example, in which case the measurement of risk is the expected number

of events in a fixed time period

Let U(8) = P r ( 0 I 8) be the cdf of 0, where 0 is the risk parameter,

which is viewed as a random variable Then U(8) represents the probability that, when a value of 0 is selected (e.g., a driver is included in our sample), the value of the risk parameter does not exceed 8 Let u(8) be the pf or pdf

of 0 Then

(5.22)

MIXED FREQUENCY MODELS 143

is the unconditional pgf of the number of events (where the formula selected depends on whether 0 is discrete or continuous2) The corresponding proba-

bilities are denoted by

The mixing distribution denoted by U(0) may be of the discrete or contin- uous type or even a combination of discrete and continuous types Discrete mixtures are mixtures of distributions where the mixing function is of the

discrete type Similarly, continuous mixtures are mixtures of distributions

where the mixing function is of the continuous type This phenomenon of mixing was introduced for continuous mixtures of severity distributions in Section 4.7.5 and for finite discrete mixtures in Section 4.5.2

It should be noted that the mixing distribution is unobservable because the data are drawn from the mixed distribution

Example 5.23 Demonstrate that the zero-modified distributions may be cre- ated by using a two-point mixture

Suppose

P ( z ) = p 1 + (1 -p)Pz(z)

This is a (discrete) two-point mixture of a degenerate distribution that places all probability at zero and a distribution with pgf Pz(z) From formula (5.12), this is also a compound Bernoulli distribution 0

Many mixed models can be constructed beginning with a simple distribu- tion Two examples are given here

Example 5.24 Determine the pf for a mixed binomial with a beta mixing dis- tribution This distribution is called binomial-beta, negative hypergeometric,

or Polya-Eggenberger

The beta distribution has pdf

'We could have written the more general P ( z ) = SP(zlO)dU(O), which would include

situations where 0 has a distribution that is partly continuous and partly discrete