dist.Skew.Discrete.Laplace {LaplacesDemon} R Documentation

## Skew Discrete Laplace Distribution: Univariate

### Description

These functions provide the density, distribution function, quantile function, and random generation for the univariate, skew discrete Laplace distribution with parameters p and q.

### Usage

dsdlaplace(x, p, q, log=FALSE)
psdlaplace(x, p, q)
qsdlaplace(prob, p, q)
rsdlaplace(n, p, q)


### Arguments

 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 log=TRUE, then the logarithm of the density is returned.

### Details

• Application: Discrete Univariate

• Density 1: p(\theta) = \frac{(1-p)(1-q)}{1-pq}p^\theta; \theta=0,1,2,3,\dots

• Density 2: p(\theta) = \frac{(1-p)(1-q)}{1-pq}q^{|\theta|}; x=0,-1,-2,-3,\dots

• Inventor: Kozubowski, T.J. and Inusah, S. (2006)

• Notation 1: \theta \sim \mathcal{DL}(p, q)

• Notation 2: p(\theta) = \mathcal{DL}(\theta | p, q)

• Parameter 1: p \in [0,1]

• Parameter 2: q \in [0,1]

• Mean 1: E(\theta) = \frac{1}{1-p}-\frac{1}{1-q}=\frac{p}{1-p}-\frac{q}{1-q}

• Mean 2: E(|\theta|) = \frac{q(1-p)^2+p(1-q)^2}{(1-qp)(1-q)(1-p)}

• Variance: var(\theta) = \frac{1}{(1-p)^2(1-q)^2}[\frac{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.

### Value

dslaplace gives the density, pslaplace gives the distribution function, qslaplace gives the quantile function, and rslaplace generates random deviates.

### References

Kozubowski, T.J. and Inusah, S. (2006). "A Skew Laplace Distribution on Integers". AISM, 58, p. 555–571.

### See Also

dalaplace, dexp, dlaplace, dlaplacep, and dslaplace.

### Examples

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"))


[Package LaplacesDemon version 16.1.6 Index]