VGAMdata (version 1.1-12)

Oiposbinom: One-Inflated Positive Binomial Distribution

Description

Density, distribution function, quantile function and random generation for the one-inflated positive binomial distribution with parameter pstr1.

Usage

doiposbinom(x, size, prob, pstr1 = 0, log = FALSE)
poiposbinom(q, size, prob, pstr1 = 0)
qoiposbinom(p, size, prob, pstr1 = 0)
roiposbinom(n, size, prob, pstr1 = 0)

Value

doiposbinom gives the density,

poiposbinom gives the distribution function,

qoiposbinom gives the quantile function, and

roiposbinom generates random deviates.

Arguments

x, p, q, n

Same as Posbinom.

size, prob

Same as Posbinom.

pstr1

Probability of a structural one (i.e., ignoring the positive binomial distribution), called \(\phi\). The default value of \(\phi = 0\) corresponds to the response having a positive binomial distribution. However, pstr1 can also be negative, in which case it ceases its interpretation as a probability, and this is known as one-deflation.

log

Logical. Return the logarithm of the answer?

Author

T. W. Yee

Details

The probability function of \(Y\) is 1 with probability \(\phi\), and \(PosBinomial(size, prob)\) with probability \(1-\phi\). Thus $$P(Y=1) =\phi + (1-\phi) P(W=1)$$ where \(W\) is distributed as a positive \(binomial(size, prob)\) random variable.

See Also

posbinomial, dbinom, binomialff.

Examples

size <- 10; prob <- 0.2; pstr1 <- 0.4; x <- (-1):size
(ii <- doiposbinom(x, size, prob, pstr1 = pstr1))
table(roiposbinom(100, size, prob, pstr1 = pstr1))
round(doiposbinom(x, size, prob, pstr1 = pstr1) * 100)  # Similar?

if (FALSE)  x <- 0:size
par(mfrow = c(2, 1))  # One-Inflated Positive Binomial
barplot(rbind(doiposbinom(x, size, prob, pstr1 = pstr1),
              dposbinom(x, size, prob)),
   beside = TRUE, col = c("blue", "orange"),
   main = paste0("OIPB(", size, ",", prob, ", pstr1 = ", pstr1,
     ") (blue) vs PosBinomial(", size, ",", prob, ") (orange)"),
   names.arg = as.character(x))

# Zero-deflated Pos Binomial:
def.limit <- -dposbinom(1, size, prob) / (1 - dposbinom(1, size, prob))
def.limit <- size * prob / (1 + (size-1) * prob-1 / (1-prob)^(size-1))
newpstr1 <- round(def.limit, 3) + 0.001  # A little from the boundary
barplot(rbind(doiposbinom(x, size, prob, pstr1 = newpstr1),
                dposbinom(x, size, prob)),
   beside = TRUE, col = c("blue","orange"),
   main = paste0("ODPB(", size, ",", prob, ", pstr1 = ", newpstr1,
      ") (blue) vs PosBinomial(", size, ",", prob, ") (orange)"),
   names.arg = as.character(x))