margmodel {weightedScores}

DENSITY AND CDF OF THE UNIVARIATE MARGINAL DISTRIBUTION

Description

Density and cdf of the univariate marginal distribution.

Usage

dmargmodel(y,mu,gam,invgam,margmodel)
pmargmodel(y,mu,gam,invgam,margmodel)
dmargmodel.ord(y,mu,gam,link)
pmargmodel.ord(y,mu,gam,link)

Arguments

y

Vector of (non-negative integer) quantiles.

mu

The parameter \mu of the univariate distribution.

gam

The parameter(s) \gamma that are not regression parameters. \gamma is NULL for Poisson and Bernoulli distribution.

invgam

The inverse of parameter \gamma of negative binomial distribution.

margmodel

Indicates the marginal model. Choices are “poisson” for Poisson, “bernoulli” for Bernoulli, and “nb1” , “nb2” for the NB1 and NB2 parametrization of negative binomial in Cameron and Trivedi (1998). See details.

link

The link function. Choices are “logit” for the logit link function, and “probit” for the probit link function.

Details

Negative binomial distribution NB(\tau,\xi) allows for overdispersion and its probability mass function (pmf) is given by

f(y;\tau,\xi)=\frac{\Gamma(\tau+y)}{\Gamma(\tau)\; y!} \frac{\xi^y}{(1+\xi)^{\tau + y}},\quad \begin{matrix} y=0,1,2,\ldots, \\ \tau>0,\; \xi>0,\end{matrix}

with mean \mu=\tau\,\xi=\exp(\beta^T x) and variance \tau\,\xi\,(1+\xi).

Cameron and Trivedi (1998) present the NBk parametrization where \tau=\mu^{2-k}\gamma^{-1} and \xi=\mu^{k-1}\gamma, 1\le k\le 2. In this function we use the NB1 parametrization (\tau=\mu\gamma^{-1},\; \xi=\gamma), and the NB2 parametrization (\tau=\gamma^{-1},\; \xi=\mu\gamma); the latter is the same as in Lawless (1987).

margmodel.ord is a variant of the code for ordinal (probit and logistic) model. In this case, the response Y is assumed to have density

f_1(y;\nu,\gamma)=F(\alpha_{y}+\nu)-F(\alpha_{y-1}+\nu),

where \nu=x\beta is a function of x and the p-dimensional regression vector \beta, and \gamma=(\alpha_1,\ldots,\alpha_{K-1}) is the $q$-dimensional vector of the univariate cutpoints (q=K-1). Note that F normal leads to the probit model and F logistic leads to the cumulative logit model for ordinal response.

Value

The density and cdf of the univariate distribution.

Author(s)

Aristidis K. Nikoloulopoulos [e-mail]
Harry Joe [e-mail]

References

Cameron, A. C. and Trivedi, P. K. (1998) Regression Analysis of Count Data. Cambridge: Cambridge University Press.

Lawless, J. F. (1987) Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics, 15, 209–225.

Examples

y<-3
gam<-2.5
invgam<-1/2.5
mu<-0.5
margmodel<-"nb2"
dmargmodel(y,mu,gam,invgam,margmodel)
pmargmodel(y,mu,gam,invgam,margmodel)
link="probit"
dmargmodel.ord(y,mu,gam,link)
pmargmodel.ord(y,mu,gam,link)