Mixed Poisson Distributions

ADVANCED DISCRETE DISTRIBUTIONS

7.3 Mixed-Frequency Distributions

7.3.2 Mixed Poisson Distributions

If we let𝑝_𝑘(𝜃)in (7.13) have the Poisson distribution, this leads to a class of distributions with useful properties. A simple example of a Poisson mixture is the two-point mixture.

EXAMPLE 7.14

Suppose that drivers can be classified as “good drivers” and “bad drivers,” each group with its own Poisson distribution. Determine the pf for this model and fit it to the data from Example 12.5. This model and its application to the data set are from Tr¨obliger [121].

From (7.13) the pf is

𝑝_𝑘=𝑝𝑒^−𝜆1𝜆^𝑘₁

𝑘! + (1 −𝑝)𝑒^−𝜆2𝜆^𝑘₂ 𝑘! .

The maximum likelihood estimates¹were calculated by Tr¨obliger to be ̂𝑝= 0.94, ̂𝜆1= 0.11, and ̂𝜆2 = 0.70. This means that about94%of drivers were “good” with a risk of𝜆1= 0.11expected accidents per year and6%were “bad” with a risk of𝜆2= 0.70 expected accidents per year. Note that it is not possible to return to the data set and

identify which were the bad drivers. □

This example illustrates two important points about finite mixtures. First, the model is probably oversimplified in the sense that risks (e.g. drivers) probably exhibit a continuum of risk levels rather than just two. The second point is that finite mixture models have a lot of parameters to be estimated. The simple two-point Poisson mixture has three parameters.

Increasing the number of distributions in the mixture to𝑟will then involve𝑟− 1mixing parameters in addition to the total number of parameters in the𝑟component distributions.

Consequently, continuous mixtures are frequently preferred.

The class of mixed Poisson distributions has some interesting properties that are developed here. Let𝑃(𝑧)be the pgf of a mixed Poisson distribution with arbitrary mixing

1Maximum likelihood estimation is discussed in Chapter 11.

distribution𝑈(𝜃). Then (with formulas given for the continuous case), by introducing a scale parameter𝜆, we have

𝑃(𝑧) =

∫ 𝑒^{𝜆𝜃(𝑧−1)}𝑢(𝜃)𝑑𝜃=

∫

[𝑒^{𝜆(𝑧−1)}]𝜃𝑢(𝜃)𝑑𝜃

=𝐸{[

𝑒^{𝜆(𝑧−1)}]_𝜃}

=𝑀_Θ[𝜆(𝑧− 1)], (7.14) where𝑀Θ(𝑧)is the mgf of the mixing distribution.

Therefore,𝑃^′(𝑧) =𝜆𝑀_Θ^′[𝜆(𝑧− 1)]and with𝑧= 1we obtain E(𝑁) =𝜆E(Θ), where 𝑁 has the mixed Poisson distribution. Also, 𝑃^′′(𝑧) = 𝜆²𝑀_Θ^′′[𝜆(𝑧− 1)], implying that E[𝑁(𝑁− 1)] =𝜆²E(Θ²)and, therefore,

Var(𝑁) =E[𝑁(𝑁− 1)] +E(𝑁) − [E(𝑁)]²

=𝜆²E(Θ²) +E(𝑁) −𝜆²[E(Θ)]²

=𝜆²Var(Θ) +E(𝑁)

>E(𝑁),

and thus for mixed Poisson distributions the variance is always greater than the mean.

Most continuous distributions in this book involve a scale parameter. This means that scale changes to distributions do not cause a change in the form of the distribution, but only in the value of its scale parameter. For the mixed Poisson distribution, with pgf (7.14), any change in𝜆is equivalent to a change in the scale parameter of the mixing distribution.

Hence, it may be convenient to simply set𝜆= 1where a mixing distribution with a scale parameter is used.

Douglas [29] proves that for any mixed Poisson distribution, the mixing distribution is unique. This means that two different mixing distributions cannot lead to the same mixed Poisson distribution and this allows us to identify the mixing distribution in some cases.

There is also an important connection between mixed Poisson distributions and compound Poisson distributions.

Definition 7.6 A distribution is said to be infinitely divisible if for all values of 𝑛 = 1,2,3,…its characteristic function𝜑(𝑧)can be written as

𝜑(𝑧) = [𝜑_𝑛(𝑧)]^𝑛,

where𝜑_𝑛(𝑧)is the characteristic function of some random variable.

In other words, taking the (1∕𝑛)th power of the characteristic function still results in a characteristic function. The characteristic function is defined as follows.

Definition 7.7 Thecharacteristic functionof a random variable𝑋is 𝜑_𝑋(𝑧) =E(𝑒^𝑖𝑧𝑋) =E(cos𝑧𝑋+𝑖sin𝑧𝑋), where𝑖=√

−1.

In Definition 7.6, “characteristic function” could have been replaced by “moment generating function” or “probability generating function,” or some other transform. That

is, if the definition is satisfied for one of these transforms, it will be satisfied for all others that exist for the particular random variable. We choose the characteristic function because it exists for all distributions, while the moment generating function does not exist for some distributions with heavy tails. Because many earlier results involved probability generating functions, it is useful to note the relationship between it and the characteristic function.

Theorem 7.8 If the probability generating function exists for a random variable𝑋, then 𝑃_𝑋(𝑧) =𝜑(−𝑖ln𝑧)and𝜑_𝑋(𝑧) =𝑃(𝑒^𝑖𝑧).

Proof:

𝑃_𝑋(𝑧) =E(𝑧^𝑋) =E(𝑒^𝑋ln𝑧) =E[𝑒^{−𝑖(𝑖ln𝑧)𝑋}] =𝜑_𝑋(−𝑖ln𝑧) and

𝜑_𝑋(𝑧) =E(𝑒^𝑖𝑧𝑋) =E[(𝑒^𝑖𝑧)^𝑋] =𝑃_𝑋(𝑒^𝑖𝑧). □ The following distributions, among others, are infinitely divisible: normal, gamma, Poisson, and negative binomial. The binomial distribution is not infinitely divisible because the exponent𝑚in its pgf must take on integer values. Dividing𝑚by𝑛 = 1,2,3,…will result in nonintegral values. In fact, no distributions with a finite range of support (the range over which positive probabilities exist) can be infinitely divisible. Now to the important result.

Theorem 7.9 Suppose that𝑃(𝑧)is a mixed Poisson pgf with an infinitely divisible mixing distribution. Then,𝑃(𝑧)is also a compound Poisson pgf and may be expressed as

𝑃(𝑧) =𝑒^{𝜆[𝑃2(𝑧)−1]},

where𝑃2(𝑧)is a pgf. If we insist that𝑃2(0) = 0, then𝑃2(𝑧)is unique.

A proof can be found in Feller [37, Chapter 12]. If we choose any infinitely divisible mixing distribution, the corresponding mixed Poisson distribution can be equivalently described as a compound Poisson distribution. For some distributions, this is a distinct advantage when carrying out numerical work, because the recursive formula (7.5) can be used in evaluating the probabilities once the secondary distribution is identified. For most cases, this identification is easily carried out. A second advantage is that, because the same distribution can be motivated in two different ways, a specific explanation is not required in order to use it. Conversely, the fact that one of these models fits well does not imply that it is the result of mixing or compounding. For example, the fact that claims follow a negative binomial distribution does not necessarily imply that individuals have the Poisson distribution and the Poisson parameter has a gamma distribution.

To obtain further insight into these results, we remark that if a counting distribution with pgf𝑃(𝑧) =∑∞

𝑛=0𝑝_𝑛𝑧^𝑛is known to be of compound Poisson form (or, equivalently, is an infinitely divisible pgf), then the quantities𝜆and𝑃2(𝑧)in Theorem 7.9 may be expressed in terms of𝑃(𝑧). Because𝑃2(0) = 0, it follows that𝑃(0) =𝑝0=𝑒^−𝜆or, equivalently,

𝜆= − ln𝑃(0). (7.15)

Thus, using (7.15),

𝑃2(𝑧) = 1 +1

𝜆ln𝑃(𝑧) = ln𝑃(0) − ln𝑃(𝑧)

ln𝑃(0) . (7.16)

The following examples illustrate the use of these ideas.

EXAMPLE 7.15

Use the preceding results and (7.14) to express the negative binomial distribution in both mixed Poisson and compound Poisson form.

The moment generating function of the gamma distribution with pdf denoted by 𝑢(𝜃)is (from Example 3.7 with𝛼replaced by𝑟and𝜃replaced by𝛽)

𝑀Θ(𝑡) = (1 −𝛽𝑡)^−𝑟=

∫

∞ 0

𝑒^𝑡𝜃𝑢(𝜃)𝑑𝜃, 𝑡 <1∕𝛽.

This is clearly infinitely divisible because[

𝑀_Θ(𝑡)]1∕𝑛is the mgf of another gamma distribution with𝑟replaced by𝑟∕𝑛. Thus, using (7.14) with𝜆= 1yields the negative binomial pgf

𝑃(𝑧) =𝑀Θ(𝑧− 1) =

∫

∞ 0

𝑒^{𝜃(𝑧−1)}𝑢(𝜃)𝑑𝜃= [1 −𝛽(𝑧− 1)]^−𝑟.

Because the gamma mixing distribution is infinitely divisible, Theorem 7.9 guar- antees that the negative binomial distribution is also of compound Poisson form, in agreement with Example 7.5. The identification of the Poisson parameter𝜆and the secondary distribution in Example 7.5, although algebraically correct, does not provide as much insight as in the present discussion. In particular, from (7.15) we find directly that

𝜆=𝑟ln(1 +𝛽) and, from (7.16),

𝑃2(𝑧) = −𝑟ln(1 +𝛽) +𝑟ln[1 −𝛽(𝑧− 1)]

−𝑟ln(1 +𝛽)

= ln

(1+𝛽−𝛽𝑧 1+𝛽

)

ln(1 +𝛽)⁻¹

= ln

( 1 − ^𝛽

1+𝛽𝑧) ln

( 1 − ^𝛽

1+𝛽

) ,

the logarithmic series pdf as before. □

EXAMPLE 7.16

Show that a mixed Poisson with an inverse Gaussian mixing distribution is the same as a Poisson–ETNB distribution with𝑟= −0.5.

The inverse Gaussian distribution is described in Appendix A. It has pdf 𝑓(𝑥) =

( 𝜃 2𝜋𝑥³

)1∕2

exp [

− 𝜃 2𝑥

(𝑥−𝜇 𝜇

)2]

, 𝑥 >0,

and mgf

𝑀(𝑡) =

∫

∞ 0

𝑒^𝑡𝑥𝑓(𝑥)𝑑𝑥= exp [𝜃

𝜇 (

1 −

√ 1 −2𝜇²

𝜃 𝑡 )]

,

where𝜃 >0and𝜇 >0are parameters. Note that [𝑀(𝑡)]^1∕𝑛= exp

[ 𝜃 𝑛𝜇

( 1 −

√ 1 −2𝜇²

𝜃 𝑡 )]

= exp

⎧⎪

⎨⎪

⎩ 𝜃∕𝑛²

𝜇∕𝑛

⎡⎢

⎢⎣ 1 −

√

1 −2(𝜇∕𝑛)² (𝜃∕𝑛²) 𝑡⎤

⎥⎥

⎦

⎫⎪

⎬⎪

⎭ .

This is the mgf of an inverse Gaussian distribution with𝜃replaced by𝜃∕𝑛²and𝜇by 𝜇∕𝑛, and thus the inverse Gaussian distribution is infinitely divisible.

Hence, by Theorem 7.9, the Poisson mixed over the inverse Gaussian distribution is also compound Poisson. Its pgf is then, from (7.14) with𝜆= 1,

𝑃(𝑧) =𝑀(𝑧− 1) = exp {𝜃

𝜇 [

1 −

√ 1 −2𝜇²

𝜃 (𝑧− 1) ]}

,

which may be represented, using (7.15) and (7.16) in the compound Poisson form of Theorem 7.9 with

𝜆= − ln𝑃(0) = 𝜃 𝜇

(√

1 +2𝜇² 𝜃 − 1

) ,

and

𝑃2(𝑧) =

𝜃 𝜇

( 1 −

√ 1 + ^2𝜇2

𝜃

) + ^𝜃

𝜇

[√

1 −^2𝜇2

𝜃 (𝑧− 1) − 1 ]

𝜇𝜃

( 1 −

√ 1 +^2𝜇_𝜃²

)

=

√ 1 −^2𝜇2

𝜃 (𝑧− 1) −

√ 1 +^2𝜇2

𝜃

1 −

√ 1 +^2𝜇2

𝜃

.

We recognize that𝑃2(𝑧)is the pgf of an extended truncated negative binomial distribution with𝑟= −1∕2and𝛽 = 2𝜇²∕𝜃. Unlike the negative binomial distribution, which is itself a member of the(𝑎, 𝑏,0)class, the compound Poisson representation is of more use for computational purposes than the original mixed Poisson formulation.□ It is not difficult to see that, if𝑢(𝜃)is the pf for any discrete random variable with pgf𝑃Θ(𝑧), then the pgf of the mixed Poisson distribution is𝑃Θ

[exp^{𝜆(𝑧−1)}]

, a compound distribution with a Poisson secondary distribution.

Table 7.2 Pairs of compound and mixed Poisson distributions.

Compound secondary Mixing

Name distribution distribution

Negative binomial Logarithmic Gamma

Neyman–Type A Poisson Poisson

Poisson–inverse Gaussian ETNB (𝑟= −0.5) Inverse Gaussian

EXAMPLE 7.17

Demonstrate that the Neyman Type A distribution can be obtained by mixing.

If in (7.14) the mixing distribution has pgf 𝑃Θ(𝑧) =𝑒^{𝜇(𝑧−1)}, then the mixed Poisson distribution has pgf

𝑃(𝑧) = exp{𝜇[𝑒^{𝜆(𝑧−1)}− 1]},

the pgf of a compound Poisson with a Poisson secondary distribution, that is, the

Neyman Type A distribution. □

A further interesting result obtained by Holgate [57] is that, if a mixing distribution is absolutely continuous and unimodal, then the resulting mixed Poisson distribution is also unimodal. Multimodality can occur when discrete mixing functions are used. For example, the Neyman Type A distribution can have more than one mode. You should try this calculation for various combinations of the two parameters. The relationships between mixed and compound Poisson distributions are given in Table 7.2.

In this chapter, we focus on distributions that are easily handled computationally.

Although many other discrete distributions are available, we believe that those discussed form a sufficiently rich class for most problems.