BNSP (version 1.0.2)

dnorm.pois: Bivariate Normal-Poisson distribution

Description

Normal-Poisson probability density function.

Usage

dnorm.pois(x,y,mu,Sigma,rate,E)

Arguments

x

Real valued sample from a univariate Normal distribution.

y

Integer valued sample from a univariate Poisson distribution.

mu

Two-dimentional vector with means, denoted as $\mu$ below.

Sigma

Two-by-two covariance matrix, denoted as $\Sigma$ below.

rate

Mean of the Poisson variable, denoted as $\lambda$ below.

E

Offset term, denoted as $E$ below.

Value

• Function dnorm.pois returns the joint probability density function of correlated Normal and Poisson random variables $$f(x,y|\theta) = \int_{c_{y-1}}^{c_{y}} N(x,y^{*}|\mu,\Sigma) dy^{*},$$ where $y^*$ denotes a continuous random variable that determines the count according to the rule $$Y = y \iff c_{y-1} < y^* < c_{y}.$$ Cut-points $c_y$ are defined by $$c_{y} = \Phi^{-1}(F(y;E\lambda)),$$ where $\Phi()$ is the Normal cdf, $F()$ the Poisson cdf, $E$ denotes an offset term and $\lambda$ is the mean of the Poisson. Further, $\mu$ and $\Sigma$ denote the mean vector and covariance matrix of $(x,y^{*})$.

  The integral is evaluated using the univariate Normal cdf $$f(y,x|\theta) = N(x|\mu_{1},\sqrt\Sigma_{11}){\Phi(\frac{c_y-E(y^*|x)}{sd(y^*|x)})-\Phi(\frac{c_{y-1}-E(y^*|x)}{sd(y^*|x)})},$$ where $E(y^*|x) = \mu_{2}+\Sigma_{12}(x-\mu_1)/\Sigma_{11}$ and $sd(y^*|x) = \sqrt{\Sigma_{22}-\Sigma_{12}^2 / \Sigma_{11}}$.

Examples

#When the covariance matrix is diagonal dnorm.pois is equal to the product of dnorm and dpois
mu<-c(0,0)
cov.mat<-matrix(c(1,0.0,0.0,1),ncol=2,nrow=2)
dnorm.pois(0,5,mu=mu,Sigma=cov.mat,rate=3,E=2)
dnorm(0,0,1)*dpois(5,6)
#Otherwise not equal
mu<-c(0,0)
cov.mat<-matrix(c(1,-0.8,-0.8,1),ncol=2,nrow=2)
dnorm.pois(0,5,mu=mu,Sigma=cov.mat,rate=3,E=2)