R: One-Inflated Zeta Distribution

Oizeta {VGAM} | R Documentation

## One-Inflated Zeta Distribution

### Description

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

### Usage

    doizeta(x, shape, pstr1 = 0, log = FALSE)
    poizeta(q, shape, pstr1 = 0)
    qoizeta(p, shape, pstr1 = 0)
    roizeta(n, shape, pstr1 = 0)

### Arguments

`x, q, p, n` | Same as `Uniform`.
`shape` | Vector of positive shape parameters.
`pstr1` | Probability of a structural one (i.e., ignoring the zeta distribution), called phi. The default value of phi = 0 corresponds to the response having an ordinary zeta distribution.
`log` | Same as `Uniform`.

### Details

The probability function of Y is 1 with probability phi, and Zeta(shape) with probability 1-phi. Thus

P(Y=1) = phi + (1-phi) * P(W=1)

where W is distributed as a zeta(shape) random variable.

### Value

`doizeta` gives the density, `poizeta` gives the distribution function, `qoizeta` gives the quantile function, and `roizeta` 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 zeta distribution. Here, `pstr1` is also permitted to lie in the interval `[-dzeta(1, shape) / (1 - dzeta(1, shape)), 0]`. The resulting probability of a unit count is less than the nominal zeta value, and the use of `pstr1` to stand for the probability of a structural 1 loses its meaning. When `pstr1` equals `-dzeta(1, shape) / (1 - dzeta(1, shape))` this corresponds to the 1-truncated zeta distribution.

### Author(s)

T. W. Yee

### See Also

`Zeta`, `zetaff`. `Otzeta`,

### Examples

    shape <- 1.5; pstr1 <- 0.3; x <- (-1):7
    (ii <- doizeta(x, shape, pstr1 = pstr1))
    max(abs(poizeta(1:200, shape) -
            cumsum(1/(1:200)^(1+shape)) / zeta(shape+1)))  # Should be 0

    ## Not run:  x <- 0:10
    par(mfrow = c(2, 1))  # One-Inflated zeta
    barplot(rbind(doizeta(x, shape, pstr1 = pstr1), dzeta(x, shape)),
            beside = TRUE, col = c("blue", "orange"),
            main = paste("OIZeta(", shape, ", pstr1 = ", pstr1, ") (blue) vs",
                         " Zeta(", shape, ") (orange)", sep = ""),
            names.arg = as.character(x))

    deflat.limit <- -dzeta(1, shape) / pzeta(1, shape, lower.tail = FALSE)
    newpstr1 <- round(deflat.limit, 3) + 0.001  # Inside but near the boundary
    barplot(rbind(doizeta(x, shape, pstr1 = newpstr1),
                    dzeta(x, shape)),
            beside = TRUE, col = c("blue","orange"),
            main = paste("ODZeta(", shape, ", pstr1 = ", newpstr1, ") (blue) vs",
                         " Zeta(", shape, ") (orange)", sep = ""),
            names.arg = as.character(x))
    ## End(Not run)

[Package VGAM version 1.0-3 Index]