VGAMdata (version 1.1-12)

Oizeta: 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)

Value

doizeta gives the density,

poizeta gives the distribution function,

qoizeta gives the quantile function, and

roizeta generates random deviates.

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.

Author

T. W. Yee

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.

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

if (FALSE)  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 = paste0("OIZeta(", shape, ", pstr1 = ", pstr1,
                 ") (blue) vs Zeta(", shape, ") (orange)"),
   names.arg = as.character(x))

deflat.limit <- -dzeta(1, shape) / pzeta(1, shape, lower.tail = FALSE)
newpstr1 <- round(deflat.limit, 3) + 0.001  # Near the boundary
barplot(rbind(doizeta(x, shape, pstr1 = newpstr1),
                dzeta(x, shape)),
   beside = TRUE, col = c("blue","orange"),
   main = paste0("ODZeta(", shape, ", pstr1 = ", newpstr1,
                 ") (blue) vs Zeta(", shape, ") (orange)"),
   names.arg = as.character(x))