The Standardized Bradford Distribution

Source: R/bradford.R

Density, distribution function, quantile function, and random generation for the Standardized Bradford Distribuction with parameter equal to c, which must be greater than 0. dbradford gives the density, pbradford gives the distribution function, qbradford gives the quantile function and rnorm generates random numbers from the distribution.

dbradford(x, c = 5, log = FALSE)

pbradford(q, c = 5, lower.tail = TRUE, log.p = FALSE)

qbradford(p, c = 5, lower.tail = TRUE, log.p = FALSE)

rbradford(n, c = 5)

Arguments

x, q

A vector of quantiles. Must be between 0 and 1.
c

The parameter of the standardized bradford distribution. Must be greater than 0.
log, log.p

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

logical; if TRUE (default), probabilities are given as P(x<=X); otherwise, P(x > X).
p

A vector of probabilities.
n

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

Source

The main definition for for the Standardized Bradford Function was taken from SciPy. The rbradford function is defined using the Inverse Transform Sampling Method, with a more in depth explanation presented in the package's vignettes. All functions were built to work like the probability functions in the{stats} package.

Value

A numeric vector.

Details

If c is not specified, it assumes the default value of 5. The Standardized Bradford Distribution has density defined as $$f(x;c) = \frac{c}{log(1+c) log(1+ cx)}$$ for \(c > 0\) and \(0 \leq x \leq 1\).

Examples

set.seed(1996)

# Generate 10 random numbers with c=10
rbradford(n=10, c=10)

#>  [1] 0.44921431 0.15698560 0.02080869 0.09485555 0.26153071 0.38826027
#>  [7] 0.79429637 0.06103814 0.01713073 0.20847441

# What value satisfies P(X <= x) = 0.5 when c=13?
qbradford(p=0.5, c=13)

#> [1] 0.2108967

# Probability of P(X <=0.5) when c=10
pbradford(q=0.5, c=10)

#> [1] 0.7472217

# Density function of x=0.375 when c=100
dbradford(x=0.375, c=100)

#> [1] 0.5628028