VGAM (version 0.7-10)

zipf: Zipf Distribution Family Function

Description

Estimates the parameter of the Zipf distribution.

Usage

zipf(N=NULL, link="loge", earg=list(), init.s=NULL)

Value

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

Details

The probability function for a response $Y$ is $$P(Y=y) = y^{-s} / \sum_{i=1}^N i^{-s},\ \ \ s>0,\ \ \ y=1,2,\ldots,N,$$ where $s$ is the exponent characterizing the distribution. The mean of $Y$, which are returned as the fitted values, is $\mu = H_{N,s-1} / H_{N,s}$ where $H_{n,m}= \sum_{i=1}^n i^{-m}$ is the $n$th generalized harmonic number.

Zipf's law is an experimental law which is often applied to the study of the frequency of words in a corpus of natural language utterances. It states that the frequency of any word is inversely proportional to its rank in the frequency table. For example, "the" and "of" are first two most common words, and Zipf's law states that "the" is twice as common as "of". Many other natural phenomena conform to Zipf's law.

References

pp.526-- of Chapter 11 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.

See Also

dzipf, zetaff.

Examples

zdat = data.frame(y = 1:5, w = c(63, 14, 5, 1, 2))
fit = vglm(y ~ 1, zipf, zdat, trace=TRUE, weight=w, cri="c")
fit = vglm(y ~ 1, zipf(link=identity, init=3.4), zdat, tra=TRUE, weight=w)
fit@misc$N
(shat = Coef(fit))
with(zdat, weighted.mean(y, w))
fitted(fit, mat=FALSE)