DiscreteLaplace: Discrete Laplace distribution

In extraDistr: Additional Univariate and Multivariate Distributions

Discrete Laplace distribution

Description

Probability mass, distribution function and random generation for the discrete Laplace distribution parametrized by location and scale.

Usage

ddlaplace(x, location, scale, log = FALSE)

pdlaplace(q, location, scale, lower.tail = TRUE, log.p = FALSE)

rdlaplace(n, location, scale)

Arguments

x, q	vector of quantiles.
location	location parameter.
scale	scale parameter; 0 < scale < 1.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
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.

Details

If U \sim \mathrm{Geometric}(1-p) and V \sim \mathrm{Geometric}(1-p), then U-V \sim \mathrm{DiscreteLaplace}(p), where geometric distribution is related to discrete Laplace distribution in similar way as exponential distribution is related to Laplace distribution.

Probability mass function

f(x) = \frac{1-p}{1+p} p^{|x-\mu|}

Cumulative distribution function

F(x) = \left\{\begin{array}{ll} \frac{p^{-|x-\mu|}}{1+p} & x < 0 \\ 1 - \frac{p^{|x-\mu|+1}}{1+p} & x \ge 0 \end{array}\right.

References

Inusah, S., & Kozubowski, T.J. (2006). A discrete analogue of the Laplace distribution. Journal of statistical planning and inference, 136(3), 1090-1102.

Kotz, S., Kozubowski, T., & Podgorski, K. (2012). The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Springer Science & Business Media.

Examples

p <- 0.45
x <- rdlaplace(1e5, 0, p)
xx <- seq(-200, 200, by = 1)
plot(prop.table(table(x)))
lines(xx, ddlaplace(xx, 0, p), col = "red")
hist(pdlaplace(x, 0, p))
plot(ecdf(x))
lines(xx, pdlaplace(xx, 0, p), col = "red", type = "s")

extraDistr documentation built on May 29, 2024, 9:31 a.m.