1. Backgrounders

1.1 Moment Notation per [JKK] and [PW]

This section contains some basic definitions and notation.

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

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The r^th corrected moment, moment about the mean, or central moment is

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Define m: = E(X).

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Variance is second central moment s²: = m₂.

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CV is Ö{m₂}/m = s_X/m.

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Index of skewness is a₃(X) = Ö{b₁(X)} = m₃/m₂^3/2.

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Index of kurtosis is a₃(X) = b₂(X) = m₄/m₂².

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Corrected from uncorrected moments:

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Raw moments in terms of the central moments:

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The r^th descending factorial moment is

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Factorial moments interms of uncorrected or raw moments:

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Raw moments in terms of factorial moments

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The cumulants, or semi-invariants, are defined as the coefficients of t^r/r! in the Taylor expansion of the MGF (see below):

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Cumulants interms of the central moments:

The characteristic function is

The probability generating function is

We have

Also, since the factorial moment generating function is

Mixtures and Stopped Sum distributions per [JKK]

[JKK] write mixtures as

Examples

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A Gamma mixture of Poissons is a negative binomial.

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

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

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A Beta mixture of NBs gives the Beta-Negative Binomial. The mixture is

Write the distributions with a Ú, so: G₁(G₂(z)) corresponds to F₁ÚF₂. Note that

Theorem. Let distributions F₁, F₂ have pgf's G₁(z) = \sump_kz^k and G₂(z), where G₂(z) depends on a parameter fin such a way that

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

2. Poisson Distribution

See [JKK] Chapter 4, especially section 3.

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Parameter: q

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Pr(X = x) = exp(-q)q^x/x!.

3. Negative Binomial Distribution

See [JKK] Chapter 5.

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

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[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.

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Parameters: k and P, Q = 1+P

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or

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((1-q))^-1 we have

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Parameter: 0 < q < 1

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Pr(X = x) = aq^x / x

5. Stopped Sum Distributions

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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.

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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.

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Polya-Aeppli 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.

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Poisson-Pascal distribution, also called the generalized Polya-Aeppli 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.

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The Generalized Poisson-Pascal 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.

Beta-Negative 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 ₂F₁ which makes it very hard to deal with.

7. Generalized Poisson-Pascal Distribution, [PW]

The GPP is a Poisson stopped-sum of extended truncated Negative Binomial distributions. It is a three parameter distribution. It has PGF

Note that provided 1-(1+P)^-r = 1-p^r > 0

Special cases of the GPP include:

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r = 1 is a Poisson-Geometric

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r > 0 is a Poisson-Pascal, aka Poisson-Negative Binomial

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-1 < r < 0 is a Poisson-ETNB, and you need the zero truncation.

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r = -1/2 is a Poisson-Inverse Gaussian mixture.

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 Poisson-Pascal 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).

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PIG Parameters: m and b.

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

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

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Pr(X = x) = [( mⁿ)/ n!][( K_l+n(mb^-1Ö{1+2b}))/( K_l+n(mb^-1) )] (1+2b)^-(l+n)/2. The rth factorial moment

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

The (a,b) classes fall into two sub-groups.

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(a,b,0) distributions are supported on the non-negative integers. They are specified through a, b, and p₀.

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(a,b,1) distributions are supported on the positive integers. There are two sub-sub-classes. The zero-truncated distributions have zero probability at zero. These include the zero-truncated Poisson, logarithmic and negative binomial distributions. The zero-modified distributions are a weighting of a degenerate distribution with a zero-truncated 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.)

Data Tables

	Item	Poisson Distribution
	Mean	q
	Variance	q
	q	m
	n/a
	m3	q
	m4	3q2+q
	CV	1/Ö{q}
	Skewness	1/Ö{q}
	Kurtosis	3+1/q
	PGF G(z)	exp(q(z-1)
	MGF f(t)	exp(q(eit-1)
	Recursions
	p0	exp(-q)
	pn	pn-1q/ n

	Item	NB Distribution
	Mean	kq/p
	Variance	kq/p2
	VM View, m and v
	p	1/v
	k	m/(v-1)
	Contagion View, m and c
	p	1/(1+cm)
	k	1/c
	m3	[( kq(1+q))/( p3)]
	m4	[( 3k2q2)/( p4)]+[( kq(p2+6q))/( p4)]
	CV	1/Ö[kq]
	Skewness	[( 1+q)/( Ö[kq])]
	Kurtosis	3+[( p2+6q)/ kq]
	PGF G(z)	(p/(1-qz))r
	MGF f(t)	(p/(1-qeit))r
	Recursions
	p0	pr
	pn	pn-1 (k+n-1)q/n
	pn+1	pn (k+n)q/(n+1)

	Item	NB Distribution
	Mean	kP
	Variance	kP(1+P)
	VM View, m and v
	P	v-1
	k	m/(v-1)
	Contagion View, m and c
	P	cm
	k	1/c
	m3	kP(1+P)(1+2P)
	m4	3k2P2(1+P)2+kP(1+P)(1+6P+6P2)
	CV	((1+P)/(kP))1/2
	Skewness	[( 1+2P)/( {kP(1+P)}1/2)]
	Kurtosis	3+[( (1+6P+6P2))/( kP(1+P))]
	PGF G(z)	(1+P-Pz)-k
	MGF f(t)
	Recursion
	p0	Q-k
	pn+1	[( k+r)/( r+1)][ P/( 1+P)]pn

	Item	NB Distribution
	Mean	m
	Variance	m(b+1)
	VM View, m and v
	m	m
	b	v-1
	Contagion View, m and c
	m	m
	b	cm
	m3
	m4
	CV
	Skewness
	Kurtosis
	PGF G(z)	exp(-m/bÖ{(1+2b(1-z))}-1 )
	MGF f(t)
	Recursion
	p0	exp(-m/b(Ö{1+2b}-1)
	p1	mÖ{1+2b}p0
	pn	[( b)/( 1+2b)](2-[ 3/ n])pn-1 +[( m2)/( 1+2b)][ 1/( n(n-1))]pn-2

	Item	NB Distribution
	Mean	m
	Variance	m(b+1)
	VM View, m and v
	m	m
	b	v-1
	Contagion View, m and c
	m	m
	b	cm
	m3
	m4
	CV
	Skewness
	Kurtosis
	PGF G(z)	exp(-m/bÖ{(1+2b(1-z))}-1 )
	MGF f(t)
	Recursion
	p0	exp(-m/b(Ö{1+2b}-1)
	p1	mÖ{1+2b}p0
	pn	[( b)/( 1+2b)](2-[ 3/ n])pn-1 +[( m2)/( 1+2b)][ 1/( n(n-1))]pn-2

	Item	NB Distribution
	Mean
	Variance
	VM View, m and v
	Contagion View, m and c
	m3
	m4
	CV
	Skewness
	Kurtosis
	PGF G(z)
	MGF f(t)
	Recursion
	p0
	pn

JKK] [JKK] Johnson, Kotz and Kemp Statistical Methods for Forecasting John Wiley and Sons 1983