Zeta {VGAM}

The Zeta Distribution

Description

Density, distribution function, quantile function and random generation for the zeta distribution.

Usage

dzeta(x, shape, log = FALSE)
pzeta(q, shape, lower.tail = TRUE)
qzeta(p, shape)
rzeta(n, shape)

Arguments

x, q, p, n

Same as Poisson.

shape

The positive shape parameter s.

lower.tail, log

Same meaning as in Normal.

Details

The density function of the zeta distribution is given by

y^{-s-1} / \zeta(s+1)

where s>0, y=1,2,\ldots, and \zeta is Riemann's zeta function.

Value

dzeta gives the density, pzeta gives the distribution function, qzeta gives the quantile function, and rzeta generates random deviates.

Note

qzeta() runs slower and slower as shape approaches 0 and shape approaches 1. The VGAM family function zetaff estimates the shape parameter s.

Author(s)

T. W. Yee

References

Johnson N. L., Kotz S., and Balakrishnan N. (1993). Univariate Discrete Distributions, 2nd ed. New York: Wiley.

See Also

zeta, zetaff, Oazeta, Oizeta, Otzeta.

Examples

dzeta(1:20, shape = 2)
myshape <- 0.5
max(abs(pzeta(1:200, myshape) -
    cumsum(1/(1:200)^(1+myshape)) / zeta(myshape+1)))  # Should be 0

## Not run:  plot(1:6, dzeta(1:6, 2), type = "h", las = 1,
               col = "orange", ylab = "Probability",
 main = "zeta probability function; orange: shape = 2; blue: shape = 1")
points(0.10 + 1:6, dzeta(1:6, 1), type = "h", col = "blue") 
## End(Not run)