# The bivariate negative binomial distribution Functions for the bivariate negative binomial distribution, as generated via trivariate reduction: density, random-number generation, and moments of the log-transformed distribution. ```r 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 $nu0$, $nu$, and $nu2$, in that order. If matrix, is read by row. - `p`: Numeric vector or three-column matrix of values for Bernoulli parameters $p0$, $p1$, and $p2$, 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. ## Returns `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 $Y1$. 2. `EY2`: Post-tranformation expectation of $Y2$. 3. `VY1`: Post-tranformation variance of $Y1$. 4. `VY2`: Post-tranformation variance of $Y2$. 5. `COV`: Post-tranformation covariance of $Y1$ and $Y2$. ## Author(s) Robert M. Kirkpatrick rkirkpatrick2@vcu.edu ## See Also `dbiLGP`, `dnbinom()`, `rnbinom()` ## Examples ```r ## 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]) ```

binegbin function