DiscreteDLM (version 1.0.0)

ALD: The Asymmetric Laplacian Distribution

Description

Density, distribution function, quantile function and random generation for the Asymmetric Laplacian Distribution (ALD). Also contains random number generation for a truncated ALD. Our code here is heavily derived from the "ald" package by Galarza and Lachos (2021), adapted to vectorize the mu parameter.

Usage

dALD(x, mu = 0, sigma = 1, p = 0.5)
pALD(q, mu = 0, sigma = 1, p = 0.5)
qALD(prob, mu = 0, sigma = 1, p = 0.5)
rALD(n, mu = 0, sigma = 1, p = 0.5)
rTALD(n, upper_tail, mu = 0, sigma = 1, p = 0.5)

Value

Each function returns a vector. dALD, pALD and qALD returns the PDF, CDF and inverse CDF of the ALD respectively. rALD returns a random sample from the ALD distribution, and rTALD returns a random sample from the ALD truncated at 0.

Arguments

mu

vector of location parameters.

sigma

vector of scale parameters.

p

ALD skew parameter.

prob

vector of probabilities.

n

number of observations.

x, q

vector of quantiles.

upper_tail

Logical denoting what portion of the distribution is retained. If TRUE, only positive values are sampled, and negative if FALSE.

Author

Daniel Dempsey (). Original code from the ald package were written by Christian E. Galarza and Victor H. Lachos.

Details

These functions are based on the three parameter ALD: $$f(x|\mu, \sigma, p) = \frac{p(1-p)}{\sigma}exp\left(-\rho_p(\frac{x-\mu}{\sigma})\right)$$ where $$\rho_p(z) = z(p - I_{z<0})$$ These functions differ than the ones provided in the ald package by vectorising the mu parameter. In addition we have implemented a function to sample from the truncated ALD (rTALD), which is needed for Bayesian quantile regression. Note that truncation is fixed at zero; the user only decides whether the negative or positive axis is discarded.

Examples

set.seed( 100 )

### Vectorised input
random_ALD <- rALD( 1000, mu = runif(1000, -100, 100) )
plot( random_ALD )

### Truncated version
trunc_random_ALD <- rTALD( 1000, TRUE, mu = 2, sigma = 3, p = 0.75 )
plot( dALD(sort(trunc_random_ALD), mu = 2, sigma = 3, p = 0.75) )