Differenced Zeta Distribution Family Function

Estimates the parameter of the differenced zeta distribution.

diffzeta(start = 1, lshape = "loglink", ishape = NULL)

Arguments

lshape, ishape

Same as zetaff.

start

Smallest value of the support of the distribution. Must be a positive integer.

Details

The PMF is $$P(Y=y) = (a/y)^{s} - (a/(1+y))^{s},\ \ s>0,\ \ y=a,a+1,\ldots,$$ where \(s\) is the positive shape parameter, and \(a\) is start. According to Moreno-Sanchez et al. (2016), this model fits quite well to about 40 percent of all the English books in the Project Gutenberg data base (about 30,000 texts). Multiple responses are handled.

Value

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

References

Moreno-Sanchez, I., Font-Clos, F. and Corral, A. (2016). Large-Scale Analysis of Zipf's Law in English Texts, PLoS ONE, 11(1), 1–19.

Author

T. W. Yee

See also

Diffzeta, zetaff, zeta, zipf, zipf.

Examples

odata <- data.frame(x2 = runif(nn <- 1000))  # Artificial data
odata <- transform(odata, shape = loglink(-0.25 + x2, inv = TRUE))
odata <- transform(odata, y1 = rdiffzeta(nn, shape))
with(odata, table(y1))
#> y1
#>    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   17 
#>  574  166   76   48   20   24    7   14    9   10    2    6    6    2    3    5 
#>   18   19   20   21   24   27   28   29   30   33   35   37   39   52   53   64 
#>    3    2    2    1    1    1    1    2    1    1    1    1    1    1    1    1 
#>   99  108  127  132  284  646 2604 
#>    1    1    1    1    1    1    1 
ofit <- vglm(y1 ~ x2, diffzeta, odata, trace = TRUE)
#> Iteration 1: loglikelihood = -1662.3344
#> Iteration 2: loglikelihood = -1658.7744
#> Iteration 3: loglikelihood = -1658.7429
#> Iteration 4: loglikelihood = -1658.7429
#> Iteration 5: loglikelihood = -1658.7429
coef(ofit, matrix = TRUE)
#>             loglink(shape)
#> (Intercept)     -0.2077567
#> x2               0.8802130