The Zipf-Mandelbrot Distribution

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)

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].

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.

Value

dzipfmb gives the density, pzipfmb gives the distribution function, qzipfmb gives the quantile function, and rzipfmb generates random deviates.

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.

Author

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

Examples

aa <- 1:10
(pp <- pzipfmb(aa, shape = 0.5, start = 1))
#>  [1] 0.5000000 0.6250000 0.6875000 0.7265625 0.7539062 0.7744141 0.7905273
#>  [8] 0.8036194 0.8145294 0.8238029
cumsum(dzipfmb(aa, shape = 0.5, start = 1))  # Should be same
#>  [1] 0.5000000 0.6250000 0.6875000 0.7265625 0.7539062 0.7744141 0.7905273
#>  [8] 0.8036194 0.8145294 0.8238029
qzipfmb(pp, shape = 0.5, start = 1) - aa  # Should be  all 0s
#>  [1] 0 0 0 0 0 0 0 0 0 0

rdiffzeta(30, 0.5)
#>  [1]     357       1       4       1       1       8       1 2353761       9
#> [10]       2       1     469      20      33       1      52     207       1
#> [19]    5589       1       6       3      53       3      87       1       1
#> [28]      12       1       1

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")
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'x' in selecting a method for function 'plot': object 'x' not found
lines(x+0.1, pzipfmb(x, shape = 0.5), col = "red", lty = 3, type = "h")
#> Error: object 'x' not found
 # \dontrun{}