The continuous power-law distribution

Density and distribution function of the continuous power-law distribution, with parameters xmin and alpha.

Usage

dplcon(x, xmin, alpha, log = FALSE)

pplcon(q, xmin, alpha, lower.tail = TRUE)

rplcon(n, xmin, alpha)

Arguments

x, q

vector of quantiles. The discrete power-law distribution is defined for x > xmin

xmin

The lower bound of the power-law distribution. For the continuous power-law, xmin >= 0. for the discrete distribution, xmin > 0.

alpha

The scaling parameter: alpha > 1.

log

logical (default FALSE) if TRUE, log values are returned.

lower.tail

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

n

Number of observations. If length(n) > 1, the length is taken to be the number required.

Value

dplcon gives the density and pplcon gives the distribution function.

Note

The discrete random number generator is very inefficient

Examples

xmin = 1; alpha = 1.5
x = seq(xmin, 10, length.out=1000)
plot(x, dplcon(x, xmin, alpha), type="l")

plot(x, pplcon(x, xmin, alpha), type="l", main="Distribution function")

n = 1000
con_rns = rplcon(n, xmin, alpha)
con_rns = sort(con_rns)
p = rep(1/n, n)
#Zipfs plot
plot(con_rns, rev(cumsum(p)), log="xy", type="l")