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

## Normal-Laplace Distribution: Univariate Asymmetric

### Description

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

, scale parameter `\sigma`

, and
tail-behavior parameters `\alpha`

and `\beta`

.

### Usage

```
dnormlaplace(x, mu=0, sigma=1, alpha=1, beta=1, log=FALSE)
rnormlaplace(n, mu=0, sigma=1, alpha=1, beta=1)
```

### Arguments

`x` |
This is a vector of data. |

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

`mu` |
This is the location parameter |

`sigma` |
This is the scale parameter |

`alpha` |
This is shape parameter |

`beta` |
This is shape parameter |

`log` |
Logical. If |

### Details

Application: Continuous Univariate

Density:

`p(\theta) = \frac{\alpha\beta}{\alpha + \beta}\phi\frac{\theta - \mu}{\sigma} [R(\alpha\sigma - \frac{\theta - \mu}{\sigma}) + R(\beta\sigma + \frac{\theta - \mu}{\sigma})]`

Inventor: Reed (2006)

Notation 1:

`\theta \sim \mathrm{NL}(\mu,\sigma,\alpha,\beta)`

Notation 2:

`p(\theta) = \mathrm{NL}(\theta | \mu, \sigma, \alpha, \beta)`

Parameter 1: location parameter

`\mu`

Parameter 2: scale parameter

`\sigma > 0`

Parameter 3: shape parameter

`\alpha > 0`

Parameter 4: shape parameter

`\beta > 0`

Mean:

Variance:

Mode:

The normal-Laplace (NL) distribution is the convolution of a normal
distribution and a skew-Laplace distribution. When the NL distribution
is symmetric (when `\alpha = \beta`

), it behaves
somewhat like the normal distribution in the middle of its range,
somewhat like the Laplace distribution in its tails, and functions
generally between the normal and Laplace distributions. Skewness is
parameterized by including a skew-Laplace component. It may be applied,
for example, to the logarithmic price of a financial instrument.

Parameters `\alpha`

and `\beta`

determine the
behavior in the left and right tails, respectively. A small value
corresponds to heaviness in the corresponding tail. As
`\sigma`

approaches zero, the NL distribution approaches a
skew-Laplace distribution. As `\beta`

approaches infinity,
the NL distribution approaches a normal distribution, though it never
quite reaches it.

### Value

`dnormlaplace`

gives the density, and
`rnormlaplace`

generates random deviates.

### References

Reed, W.J. (2006). "The Normal-Laplace Distribution and Its Relatives".
In *Advances in Distribution Theory, Order Statistics and
Inference*, p. 61–74, Birkhauser, Boston.

### See Also

`dalaplace`

,
`dallaplace`

,
`daml`

,
`dlaplace`

, and
`dnorm`

### Examples

```
library(LaplacesDemon)
x <- dnormlaplace(1,0,1,0.5,2)
x <- rnormlaplace(100,0,1,0.5,2)
#Plot Probability Functions
x <- seq(from=-5, to=5, by=0.1)
plot(x, dlaplace(x,0,0.5), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dlaplace(x,0,1), type="l", col="green")
lines(x, dlaplace(x,0,2), type="l", col="blue")
legend(2, 0.9, expression(paste(mu==0, ", ", lambda==0.5),
paste(mu==0, ", ", lambda==1), paste(mu==0, ", ", lambda==2)),
lty=c(1,1,1), col=c("red","green","blue"))
```

*LaplacesDemon*version 16.1.6 Index]