bzinb {bzinb}

The bivariate zero-inflated negative binomial distribution

Description

random generation (rbzinb), maximum likelihood estimation (bzinb), and log-likelihood. (lik.bzinb) for the bivariate zero-inflated negative binomial distribution with parameters equal to (a0, a1, a2, b1, b2, p1, p2, p3, p4).

Usage

lik.bzinb(xvec, yvec, a0, a1, a2, b1, b2, p1, p2, p3, p4, param = NULL)

rbzinb(n, a0, a1, a2, b1, b2, p1, p2, p3, p4, param = NULL)

bzinb(
  xvec,
  yvec,
  initial = NULL,
  tol = 1e-08,
  maxiter = 50000,
  showFlag = FALSE,
  vcov = FALSE
)

Arguments

xvec, yvec

a pair of bzinb random vectors. nonnegative integer vectors. If not integers, they will be rounded to the nearest integers.

a0, a1, a2

shape parameters of the latent gamma variables. They must be positive.

b1, b2

scale parameters for the latent gamma variables. They must be positive.

p1, p2, p3, p4

proportions summing up to 1 (p1 + p2 + p3 + p4 = 1). p1 is the probability of both latent Poisson variables being observed. p2 is the probability of only the first Poisson variables being observed. p3 is the probability of only the second Poisson variables being observed, and p4 is the probability of both Poisson variables being dropped out.

param

a vector of parameters ((a0, a1, a2, b1, b2, p1, p2, p3, p4)). Either param or individual parameters (a0, a1, a2, b1, b2, p1, p2, p3, p4) need to be provided.

n

number of observations.

initial

starting value of param for EM algorithm, a vector of nine values.

tol

tolerance for judging convergence. tol = 1e-8 by default.

maxiter

maximum number of iterations allowed. tol = 50000 by default.

showFlag

if TRUE, the updated parameter estimates for each iteration are printed out. If a positive integer, the updated parameter estimates for iterations greater than showFlag are printed out.

vcov

if TRUE, the variance-covariance matrix and information matrix are returned.

Details

EM theoretically guarantees higher likelihood at each iteration than that of previous iterations. See Dempster, Laird, and Rubin (1977). This guarantee comes with an assumption that there is no numerical error in conditional likelihood maximization at each iteration. Small errors can cause decreasing likelihood especially when the iterations reach the point of convergence. Due to this technical error, the EM continues after it reaches the maximum likelihood point (up to 100 iterations). However, the final estimate being returned is the parameter values at the maximum likelihood.

Value

• rbzinb gives a pair of random vectors following BZINB distribution.

• bzinb gives the maximum likelihood estimates of a BZINB pair.

  • rho estimate and standard error of the underlying correlation (\rho) and (logit(\rho))

  • coefficients estimate and standard error of the BZINB parameters

  • lik log-likelihood of the maximum likelihood estimate

  • iter total number of iterations

  • info information matrix. Provided when vcov is TRUE.

  • vcov variance-covariance matrix. Provided when vcov is TRUE.

• lik.bzinb gives the log-likelihood of a set of parameters for a BZINB pair.

Author(s)

Hunyong Cho, Chuwen Liu, Jinyoung Park, and Di Wu

References

Cho, H., Preisser, J., Liu, C., and Wu, D. (In preparation), "A bivariate zero-inflated negative binomial model for identifying underlying dependence"

Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1), 1-22.

Examples

# generating a pair of random vectors
set.seed(2)
data1 <- rbzinb(n = 100, a0 = 2, a1 = 1, a2 = 1, 
                b1 = 1, b2 = 1, p1 = 0.5, p2 = 0.2, 
                p3 = 0.2, p4 = 0.1)

lik.bzinb(xvec = data1[, 1], yvec = data1[ ,2], 
          a0 = 1, a1 = 1, a2 = 1, b1 = 1, b2 = 1, 
          p1 = 0.5, p2 = 0.2, p3 = 0.2, p4 = 0.1)

bzinb(xvec = data1[,1], yvec = data1[,2], showFlag = FALSE)