# hzeta: Haight's Zeta Family Function

### Description

Estimating the parameter of Haight's zeta distribution

### Usage

 `1` ```hzeta(lshape = "loglog", ishape = NULL, nsimEIM = 100) ```

### Arguments

 `lshape` Parameter link function for the parameter, called alpha below. See `Links` for more choices. Here, a log-log link keeps the parameter greater than one, meaning the mean is finite. `ishape,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`.

T. W. Yee

### References

Pages 533–4 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.

`Hzeta`, `zeta`, `zetaff`, `loglog`, `simulate.vlm`.

### Examples

 ```1 2 3 4 5 6 7``` ```shape <- exp(exp(-0.1)) # The parameter hdata <- data.frame(y = rhzeta(n = 1000, shape)) fit <- vglm(y ~ 1, hzeta, data = hdata, trace = TRUE, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) # Useful for intercept-only models; should be same as shape c(with(hdata, mean(y)), head(fitted(fit), 1)) summary(fit) ```

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