VGAMdata (version 1.1-12)

Oizipf: 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)

Value

doizipf gives the density,

poizipf gives the distribution function,

qoizipf gives the quantile function, and

roizipf generates random deviates.

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.

Author

T. W. Yee

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.

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))

if (FALSE)  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))