Skew Discrete Laplace Distribution: Univariate
These functions provide the density, distribution function, quantile function, and random generation for the univariate, skew discrete Laplace distribution with parameters p and q.
dsdlaplace(x, p, q, log=FALSE) psdlaplace(x, p, q) qsdlaplace(prob, p, q) rsdlaplace(n, p, q)
x |
This is a vector of data. |
p |
This is a scalar or vector of parameter p in [0,1]. |
q |
This is a scalar or vector of parameter q in [0,1]. |
prob |
This is a probability scalar or vector. |
n |
This is the number of observations, which must be a positive integer that has length 1. |
log |
Logical. If |
Application: Discrete Univariate
Density 1: p(theta) = (1-p)(1-q) / (1-pq)p^theta, theta=0,1,2,3,...
Density 2: p(theta) = (1-p)(1-q) / (1-pq)q^(|theta|),; x=0,-1,-2,-3,...
Inventor: Kozubowski, T.J. and Inusah, S. (2006)
Notation 1: theta ~ DL(p, q)
Notation 2: p(theta) = DL(theta | p, q)
Parameter 1: p in [0,1]
Parameter 2: q in [0,1]
Mean 1: E(theta) = (1 / (1-p)) - (1 / (1-q)) = (p / (1-p)) - (q / (1-q))
Mean 2: E(|theta|) = (q(1-p)^2+p(1-q)^2) / ((1-qp)(1-q)(1-p))
Variance: var(theta) = (1 / ((1-p)^2(1-q)^2))[(q(1-p)^3(1+q)+p(1-q)^3(1+p)) / (1-pq) - (p-q)^2]
Mode:
This is a discrete form of the skew-Laplace distribution. The symmetric discrete Laplace distribution occurs when p=q. DL(p,0) is a geometric distribution, and DL(0,q) is a geometric distribution of non-positive integers. The distribution is degenerate when DL(0,0). Since the geometric distribution is a discrete analog of the exponential distribution, the distribution of the difference of two geometric variables is a discrete Laplace distribution.
These functions are similar to those in the DiscreteLaplace package.
dslaplace gives the density,
pslaplace gives the distribution function,
qslaplace gives the quantile function, and
rslaplace generates random deviates.
Kozubowski, T.J. and Inusah, S. (2006). "A Skew Laplace Distribution on Integers". AISM, 58, p. 555–571.
library(LaplacesDemon)
x <- dsdlaplace(1,0.5,0.5)
x <- psdlaplace(1,0.5,0.5)
x <- qsdlaplace(0.5,0.5,0.5)
x <- rsdlaplace(5,0.5,0.5)
#Plot Probability Functions
x <- c(-3:3)
plot(x, dsdlaplace(x,0.5,0.5), ylim=c(0,0.6), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dsdlaplace(x,0.3,0.6), type="l", col="green")
lines(x, dsdlaplace(x,0.9,0.1), type="l", col="blue")
legend(-2.5, 0.5, expression(paste(p==0.5, ", ", q==0.5),
paste(p==0.3, ", ", q==0.6),
paste(p==0.9, ", ", q==0.1)),
lty=c(1,1,1), col=c("red","green","blue"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.