zetaff {VGAM} | R Documentation

## Zeta Distribution Family Function

### Description

Estimates the parameter of the zeta distribution.

### Usage

    zetaff(link = "loge", earg=list(), init.p = NULL)

### Arguments

`link` | Parameter link function applied to the (positive) parameter p. See `Links` for more choices. Choosing `loglog` constrains p>1, but may fail if the maximum likelihood estimate is less than one.
`earg` | List. Extra argument for the link. See `earg` in `Links` for general information.
`init.p` | Optional initial value for the parameter p. The default is to choose an initial value internally. If converge failure occurs use this argument to input a value.

### Details

In this long tailed distribution the response must be a positive integer. The probability function for a response Y is

P(Y=y) = 1/(y^(p+1) zeta(p+1)), p>0, y=1,2,...

where zeta is Riemann's zeta function. The parameter p is positive, therefore a log link is the default. The mean of Y is mu = zeta(p)/zeta(p+1) provided p>1. The variance of Y is zeta(p-1) / zeta(p+1) - mu^2 provided p>2.

It appears that good initial values are needed for successful convergence. If convergence is not obtained, try several values ranging from values near 0 to values about 10 or more.

### Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, `rrvglm` and `vgam`.

### Note

The `zeta` function may be used to compute values of the zeta function.

### Author(s)

T. W. Yee

### References

pp.527– of Chapter 11 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.

Knight, K. (2000) Mathematical Statistics. Boca Raton: Chapman & Hall/CRC Press.

### See Also

`zeta`, `dzeta`, `hzeta`, `zipf`.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

### Examples

    y = 1:5     # Knight, p.304
    w =  c(63, 14, 5, 1, 2)
    fit = vglm(y ~ 1, zetaff, trace=TRUE, wei=w, crit="c")
    (phat = Coef(fit)) # 1.682557
    cbind(dzeta(y, phat) * sum(w), w)

    weighted.mean(y, w)
    fitted(fit, mat=FALSE)
    predict(fit)

    # MLE should satisfy the following:
    mean(log(rep(y, w))) + zeta(1+phat, deriv=1)/zeta(1+phat) # Should be 0

[Package VGAM version 0.7-9 Index]