dist.Laplace.Mixture {LaplacesDemon} R Documentation

## Mixture of Laplace Distributions

### Description

These functions provide the density, cumulative, and random generation for the mixture of univariate Laplace distributions with probability p, location \mu and scale \sigma.

### Usage

dlaplacem(x, p, location, scale, log=FALSE)
plaplacem(q, p, location, scale)
rlaplacem(n, p, location, scale)


### Arguments

 x, q This is vector of values at which the density will be evaluated. p This is a vector of length M of probabilities for M components. The sum of the vector must be one. n This is the number of observations, which must be a positive integer that has length 1. location This is a vector of length M that is the location parameter \mu. scale This is a vector of length M that is the scale parameter \sigma, which must be positive. log Logical. If TRUE, then the logarithm of the density is returned.

### Details

• Application: Continuous Univariate

• Density: p(\theta) = \sum p_i \mathcal{L}(\mu_i, \sigma_i)

• Inventor: Unknown

• Notation 1: \theta \sim \mathcal{L}(\mu, \sigma)

• Notation 2: p(\theta) = \mathcal{L}(\theta | \mu, \sigma)

• Parameter 1: location parameters \mu

• Parameter 2: scale parameters \sigma > 0

• Mean: E(\theta) = \sum p_i \mu_i

• Variance:

• Mode:

A mixture distribution is a probability distribution that is a combination of other probability distributions, and each distribution is called a mixture component, or component. A probability (or weight) exists for each component, and these probabilities sum to one. A mixture distribution (though not these functions here in particular) may contain mixture components in which each component is a different probability distribution. Mixture distributions are very flexible, and are often used to represent a complex distribution with an unknown form. When the number of mixture components is unknown, Bayesian inference is the only sensible approach to estimation.

A Laplace mixture distribution is a combination of Laplace probability distributions.

One of many applications of Laplace mixture distributions is the Laplace Mixture Model (LMM).

### Value

dlaplacem gives the density, plaplacem returns the CDF, and rlaplacem generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

ddirichlet and dlaplace.

### Examples

library(LaplacesDemon)
p <- c(0.3,0.3,0.4)
mu <- c(-5, 1, 5)
sigma <- c(1,2,1)
x <- seq(from=-10, to=10, by=0.1)
plot(x, dlaplacem(x, p, mu, sigma, log=FALSE), type="l") #Density
plot(x, plaplacem(x, p, mu, sigma), type="l") #CDF
plot(density(rlaplacem(10000, p, mu, sigma))) #Random Deviates


[Package LaplacesDemon version 16.1.6 Index]