Hzeta {VGAM} | R Documentation

## Haight's Zeta Function

### Description

Density, distribution function, quantile function and random generation for Haight's Zeta function distribution with parameter `alpha`.

### Usage

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

### Arguments

`x, q` | Vector of quantiles. For the density, it should be a vector with positive integer values in order for the probabilities to be positive.
`p` | vector of probabilities.
`n` | number of observations. A single positive integer.
`alpha` | The parameter value. Must contain positive values and is recycled to the length of `x` or `p` or `q` if necessary.
`log` | Logical. If `log=TRUE` then the logarithm of the density is returned.

### Details

The probability function is

f(x) = (2x-1)^(-alpha) - (2x+1)^(-alpha),

where alpha>0 and x=1,2,....

### Value

`dhzeta` gives the density, `phzeta` gives the distribution function, `qhzeta` gives the quantile function, and `rhzeta` generates random deviates.

### Note

Given some response data, the VGAM family function `hzeta` estimates the parameter `alpha`.

### Author(s)

T. W. Yee

### References

Page 533 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.

### See Also

`hzeta`, `zeta`, `zetaff`.

### Examples

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

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

    ## Not run:
    alpha = 1.1; x = 1:10
    plot(x, dhzeta(x, alpha=alpha), type="h", ylim=0:1, lwd=2,
         sub=paste("alpha =", alpha), las=1, col="blue", ylab="Probability",
         main="Haight's zeta: blue=density; red=distribution function")
    lines(x+0.1, phzeta(x, alpha=alpha), col="red", lty=3, lwd=2, type="h")
    ## End(Not run)

[Package VGAM version 0.7-9 Index]