binegbin {RMKdiscrete}

The bivariate negative binomial distribution

Description

Functions for the bivariate negative binomial distribution, as generated via trivariate reduction: density, random-number generation, and moments of the log-transformed distribution.

Usage

dbinegbin(y, nu, p, log=FALSE, add.carefully=FALSE)
binegbin.logMV(nu,p,const.add=1,tol=1e-14,add.carefully=FALSE)
rbinegbin(n, nu, p)

Arguments

y

Numeric vector or two-column matrix of bivariate data. If matrix, each row corresponds to an observation.

nu

Numeric vector or three-column matrix of non-negative values for index parameters \nu _0, \nu _1, and \nu _2, in that order. If matrix, is read by row.

p

Numeric vector or three-column matrix of values for Bernoulli parameters p_0, p_1, and p_2, in that order. If matrix, is read by row. Values must be on the interval (0,1].

log

Logical; should the natural log of the probability be returned? Defaults to FALSE.

add.carefully

Logical. If TRUE, the program takes extra steps to try to prevent round-off error during the addition of probabilities. Defaults to FALSE, which is recommended, since using TRUE is slower and rarely makes a noticeable difference in practice.

const.add

Numeric vector of positive constants to add to the non-negative integers before taking their natural logarithm. Defaults to 1, for the typical \log (y+1) transformation.

tol

Numeric; must be positive. When binegbin.logMV() is calculating the second moment of the log-transformed distribution, it stops when the next term in the series is smaller than tol.

n

Integer; number of observations to be randomly generated.

Details

This bivariate negative binomial distribution is constructed from three independent latent variables, in the same manner as the bivariate Lagrangian Poisson distribution.

Function dbinegbin() is the bivariate negative binomial density (PMF). Function rbinegbin() generates random draws from the bivariate negative binomial distribution, via calls to rnbinom(). Function binegbin.logMV() numerically computes the means, variances, and covariance of a bivariate LGP distribution, after it has been log transformed following addition of a positive constant.

Vectors of numeric arguments other than tol are cycled, whereas only the first element of logical and integer arguments is used.

Value

dbinegbin() returns a numeric vector of probabilities. rbinegbin() returns a matrix of random draws, which is of type 'numeric' (rather than 'integer', even though the negative binomial only has support on the non-negative integers). binegbin.logMV() returns a numeric matrix with the following five named columns:

1. EY1: Post-tranformation expectation of Y_1.

2. EY2: Post-tranformation expectation of Y_2.

3. VY1: Post-tranformation variance of Y_1.

4. VY2: Post-tranformation variance of Y_2.

5. COV: Post-tranformation covariance of Y_1 and Y_2.

Author(s)

Robert M. Kirkpatrick [e-mail]

See Also

dbiLGP, dnbinom(), rnbinom()

Examples

## The following two lines do the same thing:
dbinegbin(y=1,nu=1,p=0.9)
dbinegbin(y=c(1,1),nu=c(1,1,1),p=c(0.9,0.9,0.9))

dbinegbin(y=c(1,1,2,2,3,5),nu=c(1,1,1,2,2,2),p=0.9)
## Due to argument cycling, the above line is doing the following three steps:
dbinegbin(y=c(1,1),nu=c(1,1,1),p=c(0.9,0.9,0.9))
dbinegbin(y=c(2,2),nu=c(2,2,2),p=c(0.9,0.9,0.9))
dbinegbin(y=c(3,5),nu=c(1,1,1),p=c(0.9,0.9,0.9))

## Inputs to dbinegbin() can be matrices, too:
dbinegbin(y=matrix(c(1,1,2,2,3,5),ncol=2,byrow=TRUE),
  nu=matrix(c(1,1,1,2,2,2,1,1,1),ncol=3,byrow=TRUE),
  p=0.9)
  
## nu0 = 0 implies independence:
a <- dbinegbin(y=c(1,3),nu=c(0,1,2),p=c(0.1,0.5,0.9))
b <- dnegbin(x=1,nu=1,p=0.5) * dnegbin(x=3,nu=2,p=0.9)
a-b #<--near zero.

( y <- rbinegbin(10,nu=c(1.1,0.87,5.5),p=c(0.87,0.89,0.90)) )
dbinegbin(y=y,nu=c(1.1,0.87,5.5),p=c(0.87,0.89,0.90))
( mv <- negbinMVP(nu=c(1.1,0.87,5.5),p=c(0.87,0.89,0.90)) )
mv[1,2] #<--Covariance of this distribution
mv[1,2]+mv[2,2] #<--Marginal variance of Y1
mv[1,2]+mv[3,2] #<--Marginal variance of Y2
mv[1,2]/(sqrt(mv[1,2]+mv[2,2])*sqrt(mv[1,2]+mv[3,2])) #<--Correlation
logmv <- binegbin.logMV(nu=c(1.1,0.87,5.5),p=c(0.87,0.89,0.90))
## Log transformation nearly cuts the correlation in half:
logmv[1,5]/sqrt(logmv[1,3]*logmv[1,4])