rALD {PReMiuM}

Asymmetric Laplace Distribution

Description

Random generation and quantile function for a Three Parameter Asymmetric Laplace Distribution as defined in Koenker and Machado (1999) for quantile regression with location parameter equal to mu, scale parameter sigma and skewness parameter p..

Usage

rALD(n, mu = 0, sigma = 1, p = 0.5)
qALD(prob, mu = 0, sigma = 1, p = 0.5, lower.tail = TRUE)

Arguments

prob

Vector of probabilities.

n

Number of observations.

mu

Location parameter.

sigma

Scale parameter.

p

Skewness parameter.

lower.tail

Logical; if TRUE (default), probabilities are P[X strictly smaller than x] otherwise, P[X > x].

Details

If mu, sigma or p are not specified they assume the default values of 0, 1 and 0.5, respectively, belonging to the Symmetric Standard Laplace Distribution denoted by ALD(0,1,0.5).

As discussed in Koenker and Machado (1999) and Yu and Moyeed (2001) we say that a random variable Y is distributed as an ALD with location parameter mu, scale parameter sigma>0 and skewness parameter p in (0,1), if its probability density function (pdf) is given by

f(y|mu,sigma,p)=p(1-p)/sigma * e^(-p_p(y-mu)/sigma))

where p_p(.) is the so called check (or loss) function defined by

p_p(u)=u(p - I(u<0))

with I() denoting the usual indicator function. This distribution is denoted by ALD(mu,sigma,p) and it's p-th quantile is equal to mu.

The scale parameter sigma must be positive and non zero. The skew parameter p must be between zero and one (0<p<1).

Value

The length of the result is determined by n for rALD, and is the maximum of the lengths of the numerical arguments for the other functions dALD, pALD and qALD.

Authors

Silvia Liverani, Queen Mary University of London, UK.

Maintainer: Silvia Liverani <[e-mail]>

References

Galarza Morales, C., Lachos Davila, V., Barbosa Cabral, C., and Castro Cepero, L. (2017) Robust quantile regression using a generalized class of skewed distributions. Stat,6: 113-130 doi:10.1002/sta4.140.

Examples

is.wholenumber(4) # TRUE
is.wholenumber(3.4) # FALSE