Haight's Zeta Family Function

Estimating the parameter of Haight's zeta distribution

Usage

hzeta(lshape = "logloglink", 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 \(y=1,2,\ldots\). 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 \(\alpha \leq 1\), and the variance is infinite if \(\alpha \leq 2\).

Value

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

References

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

Author

T. W. Yee

See also

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

Examples

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")
#> Iteration 1: coefficients = -0.035437558
#> Iteration 2: coefficients = -0.12904235
#> Iteration 3: coefficients = -0.16134991
#> Iteration 4: coefficients = -0.16391286
#> Iteration 5: coefficients = -0.16391049
#> Iteration 6: coefficients = -0.1639105
#> Iteration 7: coefficients = -0.1639105
coef(fit, matrix = TRUE)
#>             logloglink(shape)
#> (Intercept)        -0.1639105
Coef(fit)  # Useful for intercept-only models; should be same as shape
#>    shape 
#> 2.336883 
c(with(hdata, mean(y)), head(fitted(fit), 1))
#> [1] 1.130000 1.133614
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = hzeta, data = hdata, trace = TRUE, 
#>     crit = "coef")
#> 
#> Coefficients: 
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept) -0.16391    0.04756  -3.446 0.000568 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Name of linear predictor: logloglink(shape) 
#> 
#> Log-likelihood: -353.3895 on 999 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 7 
#>