R: Skew Discrete Laplace Distribution: Univariate

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) = (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.

### 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 16.1.1 Index]