hzeta {VGAM} | R Documentation

## Haight's Zeta Family Function

### Description

Estimating the parameter of Haight's Zeta function.

### Usage

    hzeta(link = "loglog", earg=list(), ialpha = NULL, nsimEIM=100)

### Arguments

`link` | Parameter link function for the parameter. See `Links` for more choices. Here, a log-log link keeps the parameter greater than one, meaning the mean is finite.
`earg` | List. Extra argument for the link. See `earg` in `Links` for general information.
`ialpha` | Optional initial value for the (positive) parameter. The default is to obtain an initial value internally. Use this argument if the default fails.
`nsimEIM` | See `CommonVGAMffArguments` for more information.

### Details

The probability function is

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

where the parameter alpha>0 and y=1,2,.... The function `dhzeta` computes this probability function. The mean of Y, which is returned as fitted values, is (1-2^(-alpha))*zeta(alpha) provided alpha > 1, where zeta is Riemann's zeta function. The mean is a decreasing function of alpha. The mean is infinite if alpha <= 1, and the variance is infinite if alpha <= 2.

### Value

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

### 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`, `loglog`.

### Examples

    alpha = exp(exp(-0.1))  # The parameter
    y = rhzeta(n=1000, alpha) # Generate some hzeta random variates
    fit = vglm(y ~ 1, hzeta, trace = TRUE, crit="c")
    coef(fit, matrix=TRUE)
    Coef(fit)  # Useful for intercept-only models; should be same as alpha
    c(mean(y), fitted(fit)[1,])
    summary(fit)

[Package VGAM version 0.7-9 Index]