CCDM Procedure

Fixed Parameter-Value Distributions

Table 2 and Table 3 define the continuous and discrete probability distributions, respectively, that you can use in the SIMULATEDSYMBOL and COUNTMODEL statements. Note the following points when reading both tables:

You can use any one of the names in the Names column to specify the distribution that the Distribution column identifies. The names are not case-sensitive.
When you specify the parameter values for a distribution in the COUNTMODEL or SIMULATEDSYMBOL statement, the values must appear in the order in which the parameters are listed in the Parameters column. For example, specifying burr(3, 0.5, 1) means you are specifying , , and ; similarly, specifying negbin(0.3, 5) means you are specifying and .

The following points pertain to the continuous distributions in Table 2:

You can use a continuous distribution in the COUNTMODEL statement only if you use the SIMULATIONMODE= option to specify a simulation mode other than the collective risk mode.
The definitions of probability density function (PDF) and cumulative distribution functions (CDF) use the following mathematical notation:
- x denotes the value of the continuous random variable in Table 2.
- , wherever z is used.
- denotes the scale parameter for the continuous distributions.
- denotes the mean parameter, except for the lognormal distribution, where is the log-scale parameter ().
- is the lower incomplete gamma function.
- is the standard normal CDF.
Parameters of the inverse Gaussian (Wald) distribution are related to the parameters of the scaled inverse Gaussian distribution as and .
The parameters of the Tweedie and scaled Tweedie distribution are related to each other as described in the section Tweedie Distributions in Chapter 26, SEVSELECT Procedure. PROC SEVSELECT uses the same parameterizations.
The parameters of the Wald and scaled Wald distributions are related to each other as and . PROC SEVSELECT uses the scaled Wald parameterization because it has a scale parameter.

Table 2: Fixed Parameter-Value Continuous Distributions

Distribution	Names	Parameters
Burr’s Type XII	BurrTypeXII
	Burr
	Bur
Exponential	Exponential
	Expo
	Exp
Gamma	Gamma
	Gamm
	Gam
Generalized Pareto	GeneralizedPareto
	GPD
	GPare
Inverse Gaussian	Wald
(Wald)	IGauss
	IGau
Lognormal	Lognormal
	Logn
	Lgn
Normal	Normal
	Gauss
	Norm
Pareto (Type I)	ParetoTypeI		if ; 0 otherwise
	Pareto1
	Pare
Pareto (Type II)	ParetoTypeII
	Pareto2
	Par2
Scaled inverse	ScaledWald
Gaussian (Wald)	SWald
	SIGau
Scaled Tweedie	ScaledTweedie
	STweedie
	STwdy
Tweedie	Tweedie
	Twee
	Twdy
Uniform	Uniform	l	if ; 0 otherwise
	Unif	r	if ;
	Uni		1 if ; 0 otherwise
Weibull	Weibull
	Weib
	Wbl

The following points pertain to the discrete distributions in Table 3:

The definition of the probability mass functions (PMF) uses m to denote the value of the discrete random variable. The table does not show an explicit expression for the CDF because the CDF of each distribution is defined in the standard manner as .
PROC CCDM supports three parameterizations of the negative binomial distribution to offer flexibility of specification. The standard parameterization (Negbin) is the same as the one you specify in the SAS PDF and CDF functions. The other two are inspired by the parameterizations that the CNTSELECT procedure supports:
- The NegbinP1 parameterization allows the variance to be linear in mean. NegbinP1(, ) is equivalent to Negbin(, ).
- The NegbinP2 parameterization allows the variance to be quadratic in mean. NegbinP2(, ) is equivalent to Negbin(, ).
The logarithmic versions of the discrete distributions correspond to the distributions that the CNTSELECT procedure uses for the count regression models. The relationship of the parameters of the logarithmic versions and their nonlogarithmic counterparts is as follows:
- LogCMPoissonLambda(, ) = CMPoisson(, ). This distribution corresponds to the parameterization of the Conway-Maxwell-Poisson distribution that PROC CNTSELECT uses when you specify the DIST=CMPOISSON and PARAMETER=LAMBDA options in the MODEL statement of PROC CNTSELECT. The parameter corresponds to the _lnNu parameter that PROC CNTSELECT reports.
- LogCMPoissonMu(, ) = CMPoisson(, ). This distribution corresponds to the default parameterization of the Conway-Maxwell-Poisson distribution that PROC CNTSELECT uses, which is equivalent to specifying the DIST=CMPOISSON and PARAMETER=MU options in the MODEL statement of PROC CNTSELECT. The parameter corresponds to the _lnNu parameter that PROC CNTSELECT reports.
- LogNegbinP1(, ) = NegbinP1(, ). This distribution corresponds to the parameterization of the negative binomial distribution that PROC CNTSELECT uses when you specify the DIST=NEGBIN(P=1) option in the MODEL statement of PROC CNTSELECT.
- LogNegbinP2(, ) = NegbinP2(, ). This distribution corresponds to the parameterization of the default negative binomial distribution that PROC CNTSELECT uses when you specify the DIST=NEGBIN option in the MODEL statement of PROC CNTSELECT.
- LogPoisson() = Poisson(). This distribution corresponds to the parameterization of the Poisson distribution that PROC CNTSELECT uses when you specify the DIST=POISSON option in the MODEL statement of PROC CNTSELECT.

Table 3: Fixed Parameter-Value Discrete Distributions

Distribution	Names	Probability Mass Function (f)
Conway-Maxwell-Poisson	CMPoisson
(Standard)	ConMaxPoi	where
	CMP
Log Conway-Maxwell-Poisson	LogCMPoissonLambda
(-parameterization)	LogCMPL	where
	LCMPL
Log Conway-Maxwell-Poisson	LogCMPoissonMu
(-parameterization)	LogCMP	where
	LCMP
Log negative binomial	LogNegbinP1
(variance = )	LogNB1	where
	LNB1
Log negative binomial	LogNegbinP2
(variance = )	LogNB2	where
	LNB2
Log Poisson	LogPoisson
	LogPoi
	LPoi
Negative binomial	NegativeBinomial
(standard)	Negbin
	Negb
Negative binomial	NegbinP1
(variance = )	Negb1	where
	NB1
Negative binomial	NegbinP2
(variance = )	Negb2	where
	NB2
Poisson	Poisson
	Pois
	Poi

Last updated: November 24, 2025