Gaitnbinom.mix: Generally-Altered, -Inflated and -Truncated Negative Binomial Distribution (GAIT--NB--NB--NB mixture)

Description

Density, distribution function, quantile function and random generation for the generally-altered, -inflated and -truncated negative binomial (NB) distribution, based on mixtures of NB distributions having different support. This distribution is sometimes abbreviated as GAIT--NB--NB--NB.

Usage

dgaitnbinom.mix(x, size.p, prob.p = NULL, munb.p = NULL,
   alter = NULL, inflate = NULL, truncate = NULL, max.support = Inf,
   pobs.a = 0, pstr.i = 0, size.a = size.p, size.i = size.p,
   prob.a = prob.p, prob.i = prob.p,
   munb.a = munb.p, munb.i = munb.p, log.arg = FALSE)
pgaitnbinom.mix(q, size.p, prob.p = NULL, munb.p = NULL,
   alter = NULL, inflate = NULL, truncate = NULL, max.support = Inf,
   pobs.a = 0, pstr.i = 0, size.a = size.p, size.i = size.p,
   prob.a = prob.p, prob.i = prob.p,
   munb.a = munb.p, munb.i = munb.p)
qgaitnbinom.mix(p, size.p, prob.p = NULL, munb.p = NULL,
   alter = NULL, inflate = NULL, truncate = NULL, max.support = Inf,
   pobs.a = 0, pstr.i = 0, size.a = size.p, size.i = size.p,
   prob.a = prob.p, prob.i = prob.p,
   munb.a = munb.p, munb.i = munb.p)
rgaitnbinom.mix(n, size.p, prob.p = NULL, munb.p = NULL,
   alter = NULL, inflate = NULL, truncate = NULL, max.support = Inf,
   pobs.a = 0, pstr.i = 0, size.a = size.p, size.i = size.p,
   prob.a = prob.p, prob.i = prob.p,
   munb.a = munb.p, munb.i = munb.p)

Arguments

x, p, q, n, log.arg

Same meaning as in NegBinomial.

size.p, size.a, size.i

Same meaning as size in NegBinomial, i.e., for an ordinary NB distribution. The first is for the main parent (inner) distribution, and the outer distribution(s) (usually spikes) concern the altered and/or inflated values. Short vectors are recycled.

prob.p, prob.a, prob.i

Same meaning as prob in NegBinomial.

munb.p, munb.a, munb.i

Same meaning as mu in NegBinomial. Only one of the prob-type and munb-type arguments should be specified. Short vectors are recycled.

alter, inflate, truncate

See Gaitnbinom.mlm. The order of precedence is the same, viz. truncation first, then altering, and finally inflation. If alter and pobs.a are both of unit length then the default probability mass function (PMF) evaluated at alter is pobs.a.

pobs.a, pstr.i

See Gaitpois.mix.

max.support

Same as Gaitnbinom.mlm.

Value

dgaitnbinom.mix gives the density (PMF), pgaitnbinom.mix gives the distribution function, qgaitnbinom.mix gives the quantile function, and rgaitnbinom.mix generates random deviates.

Details

These functions are an alternative to Gaitnbinom.mlm, and are to Gaitnbinom.mlm what Gaitpois.mix are to Gaitpois.mlm. See Gaitpois.mix for parallel information. This distribution may be suitable for heaped data whose parent distribution is overdispersed with respect to the Poisson.

In the notation of Yee and Ma (2019) these functions concern the GAIT-NB-NB-NB distribution. For information on the GAIT-NB-MLM-MLM distribution see Gaitnbinom.mlm.

Examples

Run this code

# NOT RUN {
ivec <- c(5, 15, 10); avec <- ivec; size = 10; munb <- 10
max.support <- 20; pobs.a <- 0.35; xvec <- 0:max.support
# GAT-NB-NB mixture
(pmf.a <- dgaitnbinom.mix(xvec, size.p = size, munb.p = munb,
     max.support = max.support, pobs.a = pobs.a, alter = avec))
sum(pmf.a)  # Should be 1
# }
# NOT RUN {
ind4 <- match(xvec, avec, nomatch = 0) > 0  # xvec %in% avec
plot(xvec[ ind4], pmf.a[ ind4], type = "h", col = "orange", lwd = 1.1,
   las = 1, xlim = range(xvec), main = "GAT-NB-NB",
   ylim = c(0, max(pmf.a)), xlab = "y", ylab = "Probability")  # Spikes
lines(xvec[!ind4], pmf.a[!ind4], type = "h", col = "blue") 
# }
# NOT RUN {
# GIT-NB-NB mixture
pstr.i <- 0.15
(pmf.i <- dgaitnbinom.mix(xvec, size.p = size, munb.p = munb,
   max.support = max.support, pstr.i = pstr.i, inflate = ivec))
sum(pmf.i)  # Should be 1
# }
# NOT RUN {
  # Plot the components of pmf.i
plot(xvec, (1 - pstr.i) * dnbinom(xvec, size, mu  = munb), type = "h",
     col = "blue", las = 1, xlim = range(xvec),
     main = "GIT-NB-NB",  # The inner distribution
     ylim = c(0, max(pmf.i)), xlab = "y", ylab = "Probability")
spikes <- dnbinom(ivec, size, mu = munb) * pstr.i / sum(
          dnbinom(ivec, size, mu = munb))
start.pt <- dnbinom(ivec, size, mu = munb) *
            (1 - pstr.i) / pnbinom(max.support, size, mu = munb)
segments(ivec, start.pt,  # The outer distribution
         ivec, start.pt + spikes, col = "orange", lwd = 1.1) 
# }