VGAM (version 1.1-12)

Hzeta: Haight's Zeta Distribution

Description

Density, distribution function, quantile function and random generation for Haight's zeta distribution with parameter shape.

Usage

dhzeta(x, shape, log = FALSE)
phzeta(q, shape, log.p = FALSE)
qhzeta(p, shape)
rhzeta(n, shape)

Value

dhzeta gives the density,

phzeta gives the distribution function,

qhzeta gives the quantile function, and

rhzeta generates random deviates.

Arguments

x, q, p, n

Same meaning as runif.

shape

The positive shape parameter. Called \(\alpha\) below.

log,log.p

Same meaning as in pnorm or qnorm.

Author

T. W. Yee and Kai Huang

Details

The probability function is $$f(x) = (2x-1)^{(-\alpha)} - (2x+1)^{(-\alpha)},$$ where \(\alpha>0\) and \(x=1,2,\ldots\).

See Also

hzeta, zeta, zetaff, simulate.vlm.

Examples

dhzeta(1:20, 2.1)
rhzeta(20, 2.1)

round(1000 * dhzeta(1:8, 2))
table(rhzeta(1000, 2))

if (FALSE)  shape <- 1.1; x <- 1:10
plot(x, dhzeta(x, shape = shape), type = "h", ylim = 0:1,
     sub = paste("shape =", shape), las = 1, col = "blue",
     ylab = "Probability", lwd = 2,
     main = "Haight's zeta: blue = density; orange = CDF")
lines(x+0.1, phzeta(x, shape = shape), col = "orange", lty = 3, lwd = 2,
      type = "h")