Frequency Distributions
Frequency Distributions
Stephen Mildenhall
September 1999
1. Backgrounders
1.1 Moment Notation per [JKK] and [PW]
This section contains some basic definitions and notation.
 ·

The r^{th} uncorrected moment, moment about zero, raw moment
or moment about the origin is
 ·

The r^{th} corrected moment, moment about
the mean, or central moment is
m_{r}(X) = E((XE(X))^{r}). 

 ·

Define m: = E(X).

·

Variance is second central moment s^{2}: = m_{2}.

·

CV is Ö{m_{2}}/m = s_{X}/m.

·

Index of skewness is
a_{3}(X) = Ö{b_{1}(X)} = m_{3}/m_{2}^{3/2}.

·

Index of kurtosis is
a_{3}(X) = b_{2}(X) = m_{4}/m_{2}^{2}.

·

Corrected from uncorrected moments:
m_{r} = E(XE(X))^{r} = 
r å
j = 0

(1)^{r} 
æ ç
è

r
j

ö ÷
ø

m¢_{rj}m^{j}. 

In particular:


 
 
= m¢_{4}4m¢_{3}m+6m¢_{2}m^{2}3m^{4}. 

 

Note that you can do these in your head from the binomial
coefficients, remembering that m is the only tricky one!
 ·

Raw moments in terms of the central moments:


 
 
= m_{4}+4m_{3}m+6m^{2}m¢_{2} + m^{4}. 

 

 ·

The r^{th} descending factorial moment is
[PW] call these simply factorial moments and denote them m_{(r)}.
 ·

Factorial moments interms of uncorrected or raw moments:


 
 
 
= m¢_{4}6m¢_{3}+11m¢_{2}6m. 

 

 ·

Raw moments in terms of factorial moments


 
 
 
= m_{[4]}+6m_{[3]}+7m_{[2]}+m. 

 

In general we have
m¢_{r} = 
r å
j = 1

S(r,j)m¢_{[j]} 

where S(r,j) are the Stirling numbers of the second kind.
 ·

The cumulants, or semiinvariants, are defined as the
coefficients of t^{r}/r! in the Taylor expansion of the MGF (see below):
K_{X}(t) = logM_{X}(t) = 
å
 k_{r}t^{r}/t!. 

For independent X and Y, k_{r}(X+Y) = k_{r}(X)+k_{r}(Y).
 ·

Cumulants interms of the central moments:
Generating Function Notation per [JKK]
The characteristic function is
The probability generating function is
G(z) = 
å
j

P_{j}z^{j} = E(z^{X}), 

where P_{j} = Pr(X = j). Thus f(t) = G(e^{it}). The moment generating
function is M(t) = G(e^{t}). The cumulant generating function is
K(t) = lnG(e^{t}).
We have
m¢_{r} = 
d^{r}G(e^{t}) dt^{r}

ê ê
ê

t = 0

. 

Also, since the factorial moment generating function is
we have
m¢_{[r]} = 
d^{r}G(1+t) dt^{r}

ê ê
ê

t = 0

. 

Mixtures and Stopped Sum distributions per [JKK]
[JKK] write mixtures as
NB = Poisson(Q) 
Ù
Q

Gamma(a,b). 

The PGF of a mixture is the mixture of the PGF's.
Examples
 ·

A Gamma mixture of Poissons is a negative binomial.

·

An inverse Gaussian mixture of Poissons is a PIG. The
Generalized IG distribution gives Sichel's distribution.

·

A Poisson mixture of Poissons is a Neyman Type A
distribution. By Gurland it is also a Poissonstopped sum of Poisson
distributions.

·

A Beta mixture of NBs gives the BetaNegative Binomial. The
mixture is
NB = NB(k,P) 
Ù
p = Q^{1}

Beta(a,b). 

where Q = 1+P. Here p: = Q^{1} has beta distribution with pdf

p^{a1}(1p)^{b1} B(a,b)

. 

If the PGF can be written as G_{1}(G_{2}(z)) then Feller calls the
result a ``generalized'' distribution. F_{1} the generalized
distribution and F_{2} is the generalizing distribution. These are the
infinitely divisible distributions, by Levy's theorem. They are also
called stopped sum distributions.
Write the distributions with a Ú, so: G_{1}(G_{2}(z))
corresponds to F_{1}ÚF_{2}. Note that
G_{1}(G_{2}(z)) ~ F_{1} 
Ú
 F_{2} ~ Count 
Ú
 Severity. 

SayF_{1}ÚF_{2} as F_{1}stopped summedF_{2} distribution. For example
NB = Poisson 
Ú
 Logarithmic. 

Theorem. Let distributions F_{1}, F_{2} have pgf's G_{1}(z) = \sump_{k}z^{k} and G_{2}(z), where G_{2}(z) depends on a parameter fin such a way that
G_{2}(zkf) = (G_{2}(zf))^{k}. 

Then the mixed distribution represented by
has the pgf
so
F_{2} 
Ù
 F_{1} ~ F_{1} 
Ú
 F_{2}¢. 

For example, the Poisson, binomial and negative binomial distributions
all have pgf's of the required form:

æ ç
è

p 1qz

ö ÷
ø

kf

= 
æ ç
è

æ ç
è

p 1qz

ö ÷
ø

k

ö ÷
ø

f

. 

2. Poisson Distribution
See [JKK] Chapter 4, especially section 3.
 ·

Parameter: q

·

Pr(X = x) = exp(q)q^{x}/x!.
3. Negative Binomial Distribution
See [JKK] Chapter 5.
 ·

Parameters: k = r and p, q: = 1p.

·


Pr
 (X = x) = 
æ ç
è

k +x1
k1

ö ÷
ø

p^{k} q^{x} = 
G(k +x) G(k)x!

p^{k} q^{x} 

[JKK] prefer a parameterization by P and k. They write
Q = 1+P. Then p = 1/(1+P) = 1/Q. [PW] use r = k and b = P. These
give the following view.
 ·

Parameters: k and P, Q = 1+P

·


Pr
 (X = x) = 
æ ç
è

k +x1
k1

ö ÷
ø


æ ç
è

1 
P Q

ö ÷
ø

k


æ ç
è

P Q

ö ÷
ø

x



 ·

or

Pr
 (X = x) = 
æ ç
è

k +x1
k1

ö ÷
ø


æ ç
è

1 1+p

ö ÷
ø

k


æ ç
è

p 1+p

ö ÷
ø

x



4. Logarithmic Distribution
See [JKK] Chapter 7. This is a single parameter family supported on
the positive integers. The parameter is q. Letting
a = ln((1q))^{1} we have
This distribution is not easy to deal with.
 ·

Parameter: 0 < q < 1

·

Pr(X = x) = aq^{x} / x
5. Stopped Sum Distributions
 ·

Neyman Type A: Poisson sum of Poissons. Limited since ratio of
skewness of kurtosis falls in a tight range. No closed form expression
for density, but easy to use FFT methods. See
other Neyman distributions. See [JKK] Chapter 9, Section 6.

·

Thomas's Distribution is a Neyman Type A, where the summed
distribution is a shifted Poisson, ensuring that each occurrence
yeilds at least one claim. See page 392.

·

PolyaAeppli distribution is a Poisson stopped Shifted Geometric
distribution. The Geometric distribution is a NB with k = 1, so the
variance multiplier equals m+1. Could be useful for clash, but the
``number of claims per occurrence'' distribution is very
limited. Again, no closed form for probabilities but easy to estimate
using FFT. See page 378.

·

PoissonPascal distribution, also called the generalized
PolyaAeppli distribution, is a Poisson stopped sum of negative
binomial distributions. Can also be regarded as a mixture of negative
binomial (k,P)'s where k has a Poisson distribution. See page 382.

·

The Generalized PoissonPascal distribution ([PW] page 259) is a
Poisson stopped sum of truncated (at zero) negative binomial
distributions. The PGF is obvious.
Per an interesting table on p 253 of we have the following
formulae for the third moments about the mean.


 
m_{3} = 3s^{2}2m+ 
m2 m1


(s^{2}m)^{2} m


 
m_{3} = 3s^{2}2m+ 2 
(s^{2}m)^{2} m


 
m_{3} = 3s^{2}2m+ 
3 2


(s^{2}m)^{2} m


 
m_{3} = 3s^{2}2m+ 
(s^{2}m)^{2} m


 
m_{3} = 3s^{2}2m+ 
r+2 r+1


(s^{2}m)^{2} m



 

Note that r > 1 in the last line give a great deal of flexibility.
BetaNegative Binomial, a NB mixed over the variance multplier
distributed as a beta should have a lot of potential as a
distribution. However, the PGF involves _{2}F_{1} which makes it very
hard to deal with.
7. Generalized PoissonPascal
Distribution, [PW]
The GPP is a Poisson stoppedsum of extended truncated Negative Binomial
distributions. It is a three parameter distribution. It has PGF
G(z) = exp 
æ ç
è

q 
æ ç
è

(1+PPz)^{r}(1+P)^{r} 1(1+P)^{r}

1 
ö ÷
ø

ö ÷
ø

. 

Note that provided 1(1+P)^{r} = 1p^{r} > 0
G(z) = exp 
æ ç
è

q 1(1+P)^{r}

((1+PPz)^{r}1) 
ö ÷
ø



is a valid PGF for a PoissonNegative Binomial (PoissonPascal). The
condition is necessary so that the frequency is nonnegative.
Thus in the PoissonPascal case the distribution can be
regarded as a PoissonNB without zero truncation, or a PoissonZTNB,
with an adjusted primary Poisson frequency.
Special cases of the GPP include:
 ·

r = 1 is a PoissonGeometric

·

r > 0 is a PoissonPascal, aka PoissonNegative Binomial

·

1 < r < 0 is a PoissonETNB, and you need the zero truncation.

·

r = 1/2 is a PoissonInverse Gaussian mixture.
8. PIG and GPIG Distributions
The PIG is a Poisson mixed over an Inverse Gaussian distribution. The
PIG is closed under certain convolutions, see [PW]. It has a thicker
tail than the Negative Binomial distribution. It is a special case of the
generalized PoissonPascal distribution with r = 1/2.
References for this section are from [PW], Section 7.8.3.
Per page 260, the PIG is a Poisson ETNB.
Per page 261, the Poisson ETNB with 1 < r < 0 is a Poisson
mixture with a stable distribution, (see also Feller p 448, 581).
 ·

PIG Parameters: m and b.

·

See below with l = 1/2.
The Generalized Poisson inverse Gaussian distribution is also called
Sichel's distribution.
 ·

Sichel's Distribution Parameters: m and b
and l.

·

Pr(X = x) = [( m^{n})/ n!][( K_{l+n}(mb^{1}Ö{1+2b}))/( K_{l+n}(mb^{1}) )] (1+2b)^{(l+n)/2}.
The rth factorial moment
m_{[r]} = m^{r} 
K_{l+r}(m/b) K_{l}(m/b)

. 

The Bessel function used is the modified Bessel function of the third
(second according to some sources!)
kind, K_{l}(x). It is available for integral l built
into Excel, in MathFunctions as nrBesselK(n,x) for any real n and
x Î R and also in Matlab as nrBesselK, again for any n and
any x Î C.
Matlab mentions their BesselK uses a MEX interface to a Fortran
library by D. E. Amos, which are available on the
web under www.netlib.com, search for amos.
9. Recursive Classes of Distributions
The (a,b) recursion is
For (a,b,0) the recursion is valid for n = 1,2,3,.... For
(a,b,1) the recursion is valid for n = 2,3,4,....
The (a,b) classes fall into two subgroups.
 ·

(a,b,0) distributions are supported on the nonnegative
integers. They are specified through a, b, and p_{0}.

·

(a,b,1) distributions are supported on the positive
integers. There are two subsubclasses. The zerotruncated
distributions have zero probability at zero. These include the
zerotruncated Poisson, logarithmic and negative binomial
distributions. The zeromodified distributions are a weighting of a
degenerate distribution with a zerotruncated class.
For the negative binomial, there is slightly more flexibility in the
choice of parameters for the truncated distribution, so it is
sometimes called the ``extended truncated negative binomial
distribution''. Normally we have parameters r = k and b = P = q/p, with mean
rP and variance multiplier 1+P = Q. In the ETNB, we must still have
b > 0, so the apparent variance multiplier is greater than
1. However, we can have 1 < r < 0, which would translate into a
negative mean, in the usual case. Also, if r < 0 then the probability
of a zero loss is p^{r} > 1, which is also impossible, since p < 1
always. (Recall, p = 1/(1+P) = 1/vm.)
See the nice table on p 229 of for a good summary of the
options. See also page 250251 for a chart showing the relationships
between the various distributions.
Data Tables
Poisson Distribution Key Facts


 Item  Poisson Distribution 

  
 Mean  q 
 Variance  q 
  
 q  m 
 n/a  
 m_{3}  q 
 m_{4}  3q^{2}+q 
 CV  1/Ö{q} 
 Skewness  1/Ö{q} 
 Kurtosis  3+1/q 
  
 PGF G(z)  exp(q(z1) 
 MGF f(t)  exp(q(e^{it}1) 
  
 Recursions  
 p_{0}  exp(q) 
 p_{n}  p_{n1}q/ n 
  

Negative Binomial (r = k,p) Key Facts


 Item  NB Distribution 

  
 Mean  kq/p 
 Variance  kq/p^{2} 
  
 VM View, m and v  
 p  1/v 
 k  m/(v1) 
 Contagion View, m and c  
 p  1/(1+cm) 
 k  1/c 
  
 m_{3}  [( kq(1+q))/( p^{3})] 
 m_{4}  [( 3k^{2}q^{2})/( p^{4})]+[( kq(p^{2}+6q))/( p^{4})] 
 CV  1/Ö[kq] 
 Skewness  [( 1+q)/( Ö[kq])] 
 Kurtosis  3+[( p^{2}+6q)/ kq] 
  
 PGF G(z)  (p/(1qz))^{r} 
 MGF f(t)  (p/(1qe^{it}))^{r} 
  
 Recursions  
 p_{0}  p^{r} 
 p_{n}  p_{n1} (k+n1)q/n 
 p_{n+1}  p_{n} (k+n)q/(n+1) 
  

Table
Negative Binomial (k,P) Key Facts


 Item  NB Distribution 

  
 Mean  kP 
 Variance  kP(1+P) 
  
 VM View, m and v  
 P  v1 
 k  m/(v1) 
 Contagion View, m and c  
 P  cm 
 k  1/c 
  
 m_{3}  kP(1+P)(1+2P) 
 m_{4}  3k^{2}P^{2}(1+P)^{2}+kP(1+P)(1+6P+6P^{2}) 
 CV  ((1+P)/(kP))^{1/2} 
 Skewness  [( 1+2P)/( {kP(1+P)}^{1/2})] 
 Kurtosis  3+[( (1+6P+6P^{2}))/( kP(1+P))] 
  
 PGF G(z)  (1+PPz)^{k} 
 MGF f(t)  
  
 Recursion  
 p_{0}  Q^{k} 
 p_{n+1}  [( k+r)/( r+1)][ P/( 1+P)]p_{n} 
  

Table
PIG Distribution Key Facts


 Item  NB Distribution 

  
 Mean  m 
 Variance  m(b+1) 
  
 VM View, m and v  
 m  m 
 b  v1 
 Contagion View, m and c  
 m  m 
 b  cm 
  
 m_{3}  
 m_{4}  
 CV  
 Skewness  
 Kurtosis  
  
 PGF G(z)  exp(m/bÖ{(1+2b(1z))}1 ) 
 MGF f(t)  
  
 Recursion  
 p_{0}  exp(m/b(Ö{1+2b}1) 
 p_{1}  mÖ{1+2b}p_{0} 
 p_{n}  [( b)/( 1+2b)](2[ 3/ n])p_{n1} +[( m^{2})/( 1+2b)][ 1/( n(n1))]p_{n2} 
  

Table
Generalized PIG Distribution Key Facts


 Item  NB Distribution 

  
 Mean  m 
 Variance  m(b+1) 
  
 VM View, m and v  
 m  m 
 b  v1 
 Contagion View, m and c  
 m  m 
 b  cm 
  
 m_{3}  
 m_{4}  
 CV  
 Skewness  
 Kurtosis  
  
 PGF G(z)  exp(m/bÖ{(1+2b(1z))}1 ) 
 MGF f(t)  
  
 Recursion  
 p_{0}  exp(m/b(Ö{1+2b}1) 
 p_{1}  mÖ{1+2b}p_{0} 
 p_{n}  [( b)/( 1+2b)](2[ 3/ n])p_{n1} +[( m^{2})/( 1+2b)][ 1/( n(n1))]p_{n2} 
  

FILLINNAME Key Facts


 Item  NB Distribution 

  
 Mean  
 Variance  
  
 VM View, m and v  
  
  
 Contagion View, m and c  
  
  
  
 m_{3}  
 m_{4}  
 CV  
 Skewness  
 Kurtosis  
  
 PGF G(z)  
 MGF f(t)  
  
 Recursion  
 p_{0}  
 p_{n}  
  

JKK]
[JKK]
Johnson, Kotz and Kemp
Statistical Methods for Forecasting
John Wiley and Sons
1983
File translated from T_{E}X by T_{T}H, version 2.34.
On 11 Sep 1999, 17:28.