f_negbinomial: Specify a negative binomial sampling distribution

In mcmcsae: Markov Chain Monte Carlo Small Area Estimation

Specify a negative binomial sampling distribution

Description

This function can be used in the family argument of create_sampler or generate_data to specify a negative binomial sampling distribution.

Usage

f_negbinomial(
  link = "log",
  shape.vec = ~1,
  inv.shape.prior = pr_invchisq(df = 1, scale = 1),
  control = negbin_control()
)

Arguments

link	the name of a link function. Currently the only allowed link function for the negative binomial sampling distribution is "log".
shape.vec	optional formula specification of unequal shape values. The negative binomial (vector) shape parameter is then equal to this vector of shape values, multiplied by the scalar shape parameter, whose prior is specified through inv.shape.prior.
inv.shape.prior	Prior on the (scalar) reciprocal shape parameter, i.e. the overdispersion parameter. Supported prior distributions are pr_fixed with a default value of 1, pr_invchisq and pr_gig. The current default is pr_invchisq(df=1, scale=1).
control	a list with computational options. These options can be specified using function negbin_control.

Details

The negative binomial distribution with shape r and probability p has density

p(y|r, p) = {\Gamma(y + r)\over y!\Gamma(r)}(1-p)^r p^y

with mean \mu = E(y|r,p) = {rp\over 1-p} and variance V(y|r,p) = \mu(1 + \mu/r). The second term of the variance can be interpreted as overdispersion with respect to a Poisson distribution, which would correspond to the limit r \rightarrow \infty. So the reciprocal shape 1/r is an overdispersion parameter, which typically is inferred. It is assigned a default prior, which may be changed through argument inv.shape.prior.

The only supported link function is "log". Strictly speaking the relation between mean \mu and linear predictor \eta is

\log\mu = \log r + \log{p\over 1-p} = \log r + \eta

This way the likelihood function has the same form as that of logistic binomial regression, so that a Polya-Gamma data augmentation sampling algorithm can be employed. Note that the fact that the linear predictor \eta does not include \log r effectively changes the interpretation of its intercept.

Value

A family object.

References

N. Polson, J.G. Scott and J. Windle (2013). Bayesian Inference for Logistic Models Using Polya-Gamma Latent Variables. Journal of the American Statistical Association 108(504), 1339-1349.

M. Zhou and L. Carin (2015). Negative Binomial Process Count and Mixture Modeling. IEEE Transactions on Pattern Analysis and Machine Intelligence 37(2), 307-320.

mcmcsae documentation built on June 8, 2025, 10:55 a.m.