Oizipf {VGAMdata}

One-Inflated Zipf Distribution

Description

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

Usage

doizipf(x, N, shape, pstr1 = 0, log = FALSE)
poizipf(q, N, shape, pstr1 = 0)
qoizipf(p, N, shape, pstr1 = 0)
roizipf(n, N, shape, pstr1 = 0)

Arguments

x, q, p, n

Same as Uniform.

N, shape

See Zipf.

pstr1

Probability of a structural one (i.e., ignoring the Zipf distribution), called \phi. The default value of \phi = 0 corresponds to the response having an ordinary Zipf distribution.

log

Same as Uniform.

Details

The probability function of Y is 1 with probability \phi, and Zipf(N, s) with probability 1-\phi. Thus

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

where W is distributed as a Zipf(N, s) random variable. The VGAM family function oizeta estimates the two parameters of this model by Fisher scoring.

Value

doizipf gives the density, poizipf gives the distribution function, qoizipf gives the quantile function, and roizipf 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 Zipf distribution. Here, pstr1 is also permitted to lie in the interval [-dzipf(1, N, s) / (1 - dzipf(1, N, s)), 0]. The resulting probability of a unit count is less than the nominal zipf value, and the use of pstr1 to stand for the probability of a structural 1 loses its meaning. When pstr1 equals -dzipf(1, N, s) / (1 - dzipf(1, N, s)) this corresponds to the 1-truncated zipf distribution.

Author(s)

T. W. Yee

See Also

oizeta. Zipf, zipf, Oizeta.

Examples

N <- 10; shape <- 1.5; pstr1 <- 0.3; x <- (-1):N
(ii <- doizipf(x, N, shape, pstr1 = pstr1))

## Not run:  x <- 0:10
par(mfrow = c(2, 1))  # One-Inflated zipf
barplot(rbind(doizipf(x, N, shape, pstr1 = pstr1),
                dzipf(x, N, shape)),
   beside = TRUE, col = c("blue", "orange"),
   main = paste0("OIZipf(", N, ", ", shape, ", pstr1 = ", pstr1,
                 ") (blue) vs Zipf(", N, ", ", shape, ") (orange)"),
   names.arg = as.character(x))

deflat.limit <- -dzipf(1, N, shape) / (1 - dzipf(1, N, shape))
newpstr1 <- round(deflat.limit, 3) + 0.001  # Near the boundary
barplot(rbind(doizipf(x, N, shape, pstr1 = newpstr1),
                dzipf(x, N, shape)),
   beside = TRUE, col = c("blue", "orange"),
   main = paste0("ODZipf(", N, ", ", shape, ", pstr1 = ", newpstr1,
                 ") (blue) vs Zipf(", N, ", ", shape, ") (orange)"),
   names.arg = as.character(x)) 
## End(Not run)