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

## Skew-Laplace Distribution: Univariate

### Description

These functions provide the density, distribution function, quantile
function, and random generation for the univariate, skew-Laplace
distribution with location parameter `\mu`

, and two mixture
parameters: `\alpha`

and `\beta`

.

### Usage

```
dslaplace(x, mu, alpha, beta, log=FALSE)
pslaplace(q, mu, alpha, beta)
qslaplace(p, mu, alpha, beta)
rslaplace(n, mu, alpha, beta)
```

### Arguments

`x` , `q` |
These are each a vector of quantiles. |

`p` |
This is a vector of probabilities. |

`n` |
This is the number of observations, which must be a positive integer that has length 1. |

`mu` |
This is the location parameter |

`alpha` |
This is a mixture parameter |

`beta` |
This is a mixture parameter |

`log` |
Logical. If |

### Details

Application: Continuous Univariate

Density 1:

`p(\theta) = \frac{1}{\alpha + \beta} \exp(\frac{\theta - \mu}{\alpha}), \theta \le \mu`

Density 2:

`p(\theta) = \frac{1}{\alpha + \beta} \exp(\frac{\mu - \theta}{\beta}), \theta > \mu`

Inventor: Fieller, et al. (1992)

Notation 1:

`\theta \sim \mathcal{SL}(\mu, \alpha, \beta)`

Notation 2:

`p(\theta) = \mathcal{SL}(\theta | \mu, \alpha, \beta)`

Parameter 1: location parameter

`\mu`

Parameter 2: mixture parameter

`\alpha > 0`

Parameter 3: mixture parameter

`\beta > 0`

Mean:

`E(\theta) = \mu + \beta - \alpha`

Variance:

`var(\theta) = \alpha^2 + \beta^2`

Mode:

`mode(\theta) = \mu`

This is the three-parameter general skew-Laplace distribution, which is
an extension of the two-parameter central skew-Laplace distribution. The
general form allows the mode to be shifted along the real line with
parameter `\mu`

. In contrast, the central skew-Laplace has mode
zero, and may be reproduced here by setting `\mu=0`

.

The general skew-Laplace distribution is a mixture of a negative
exponential distribution with mean `\beta`

, and the negative
of an exponential distribution with mean `\alpha`

. The
weights of the positive and negative components are proportional to
their means. The distribution is symmetric when
`\alpha=\beta`

, in which case the mean is `\mu`

.

These functions are similar to those in the `HyperbolicDist`

package.

### Value

`dslaplace`

gives the density,
`pslaplace`

gives the distribution function,
`qslaplace`

gives the quantile function, and
`rslaplace`

generates random deviates.

### References

Fieller, N.J., Flenley, E.C., and Olbricht, W. (1992). "Statistics of
Particle Size Data". *Applied Statistics*, 41, p. 127–146.

### See Also

`dalaplace`

,
`dexp`

,
`dlaplace`

,
`dlaplacep`

, and
`dsdlaplace`

.

### Examples

```
library(LaplacesDemon)
x <- dslaplace(1,0,1,1)
x <- pslaplace(1,0,1,1)
x <- qslaplace(0.5,0,1,1)
x <- rslaplace(100,0,1,1)
#Plot Probability Functions
x <- seq(from=0.1, to=3, by=0.01)
plot(x, dslaplace(x,0,1,1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dslaplace(x,0,0.5,2), type="l", col="green")
lines(x, dslaplace(x,0,2,0.5), type="l", col="blue")
legend(1.5, 0.9, expression(paste(mu==0, ", ", alpha==1, ", ", beta==1),
paste(mu==0, ", ", alpha==0.5, ", ", beta==2),
paste(mu==0, ", ", alpha==2, ", ", beta==0.5)),
lty=c(1,1,1), col=c("red","green","blue"))
```

*LaplacesDemon*version 16.1.6 Index]