weightedScores (version 0.9.5.1)

margmodel: 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 Triv

link

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

Value

• The density and cdf of the univariate distribution.

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.

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.

Nikoloulopoulos, A.K. (2015b) Weighted scores estimating equations for longitudinal ordinal data. Arxiv e-prints.

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)