R: The Zeta Distribution

Zeta {VGAM} | R Documentation

## 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 p.
`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,…, 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 `p` 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)

[Package VGAM version 1.1-3 Index]