VGAM (version 1.1-12)

Zipfmb: The Zipf-Mandelbrot Distribution

Description

Density, distribution function, quantile function and random generation for the Mandelbrot distribution.

Usage

dzipfmb(x, shape, start = 1, log = FALSE)
pzipfmb(q, shape, start = 1, lower.tail = TRUE, log.p = FALSE)
qzipfmb(p, shape, start = 1)
rzipfmb(n, shape, start = 1)

Value

dzipfmb gives the density,

pzipfmb gives the distribution function,

qzipfmb gives the quantile function, and

rzipfmb generates random deviates.

Arguments

x

vector of (non-negative integer) quantiles.

q

vector of quantiles.

p

vector of probabilities.

n

number of random values to return.

shape

vector of positive shape parameter.

start

integer, the minimum value of the support of the distribution.

log, log.p

logical; if TRUE, probabilities p are given as log(p)

lower.tail

logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

Author

M. Chou, with edits by T. W. Yee.

Details

The probability mass function of the Zipf-Mandelbrot distribution is given by $$\Pr(Y=y;s) = \frac{s \; \Gamma(y_{min})}{\Gamma(y_{min}-s)} \cdot \frac{\Gamma(y-s)}{\Gamma(y+1)}$$ where \(0 \leq b < 1\) and the starting value start being by default 1.

References

Mandelbrot, B. (1961). On the theory of word frequencies and on related Markovian models of discourse. In R. Jakobson, Structure of Language and its Mathematical Aspects, pp. 190--219, Providence, RI, USA. American Mathematical Society.

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

See Also

Zipf.

Examples

aa <- 1:10
(pp <- pzipfmb(aa, shape = 0.5, start = 1))
cumsum(dzipfmb(aa, shape = 0.5, start = 1))  # Should be same
qzipfmb(pp, shape = 0.5, start = 1) - aa  # Should be  all 0s

rdiffzeta(30, 0.5)

if (FALSE) x <- 1:10
plot(x, dzipfmb(x, shape = 0.5), type = "h", ylim = 0:1,
     sub = "shape=0.5", las = 1, col = "blue", ylab = "Probability",
     main = "Zipf-Mandelbrot distribution: blue=PMF; orange=CDF")
lines(x+0.1, pzipfmb(x, shape = 0.5), col = "red", lty = 3, type = "h")