Oiposbinom {VGAM} | R Documentation

## 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)

### 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?

### 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.

### Value

`doiposbinom` gives the density, `poiposbinom` gives the distribution function, `qoiposbinom` gives the quantile function, and `roiposbinom` generates random deviates.

### Note

The argument `pstr1` is recycled to the required length, and usually has values which lie in the interval [0,1]. These functions actually allow for the zero-deflated binomial distribution. Here, `pstr1` is also permitted to lie in the interval [-A, 0] for some positive quantity A. The resulting probability of a unit value is less than the nominal positive binomial value, and the use of `pstr1` to stand for the probability of a structural 1 loses its meaning. If `pstr1` equals A then this corresponds to the 0- and 1-truncated binomial distribution.

### Author(s)

T. W. Yee

### 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)  # Should be similar

    ## Not run:  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 = paste("OIPB(", size, ",", prob, ", pstr1 = ", pstr1, ") (blue) vs",
                         " PosBinomial(", size, ",", prob, ") (orange)", sep = ""),
            names.arg = as.character(x))

    # Zero-deflated Pos Binomial
    deflat.limit <- -dposbinom(1, size, prob) / (1 - dposbinom(1, size, prob))
    deflat.limit <- size * prob / (1 + (size-1) * prob - 1 / (1-prob)^(size-1))
    newpstr1 <- round(deflat.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 = paste("ODPB(", size, ",", prob, ", pstr1 = ", newpstr1, ") (blue) vs",
                         " PosBinomial(", size, ",", prob, ") (orange)", sep = ""),
            names.arg = as.character(x))
    ## End(Not run)

[Package VGAM version 1.1-1 Index]