ald (version 1.3.1)

ALD: The Asymmetric Laplace Distribution

Description

Density, distribution function, quantile function and random generation for a Three-Parameter Asymmetric Laplace Distribution defined in Koenker and Machado (1999) useful for quantile regression with location parameter equal to mu, scale parameter sigma and skewness parameter p This is a special case of the skewed family of distributions in Galarza (2016) available in lqr::SKD.

Usage

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

Arguments

y,q

vector of quantiles.

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 \(\le\) x] otherwise, P[X > x].

Value

dALD gives the density, pALD gives the distribution function, qALD gives the quantile function, and rALD generates a random sample.

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.

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

$$f(y|\mu,\sigma,p)=\frac{p(1-p)}{\sigma}\exp {-\rho_{p}(\frac{y-\mu}{\sigma})}$$

where \(\rho_p(.)\) is the so called check (or loss) function defined by $$\rho_p(u)=u(p - I_{u<0})$$, 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).

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.

Yu, K., & Zhang, J. (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics-Theory and Methods, 34(9-10), 1867-1879.

See Also

momentsALD,likALD,mleALD

Examples

# NOT RUN {
## Let's plot an Asymmetric Laplace Distribution!

##Density
library(ald)
sseq = seq(-40,80,0.5)
dens = dALD(y=sseq,mu=50,sigma=3,p=0.75)
plot(sseq,dens,type = "l",lwd=2,col="red",xlab="x",ylab="f(x)", main="ALD Density function")

#Look that is a special case of the skewed family in Galarza (2017)
# available in lqr package, dSKD(...,sigma = 2*3,dist = "laplace")

## Distribution Function
df = pALD(q=sseq,mu=50,sigma=3,p=0.75)
plot(sseq,df,type="l",lwd=2,col="blue",xlab="x",ylab="F(x)", main="ALD Distribution function")
abline(h=1,lty=2)

##Inverse Distribution Function
prob = seq(0,1,length.out = 1000)
idf = qALD(prob=prob,mu=50,sigma=3,p=0.75)
plot(prob,idf,type="l",lwd=2,col="gray30",xlab="x",ylab=expression(F^{-1}~(x)))
title(main="ALD Inverse Distribution function")
abline(v=c(0,1),lty=2)

#Random Sample Histogram
sample = rALD(n=10000,mu=50,sigma=3,p=0.75)
hist(sample,breaks = 70,freq = FALSE,ylim=c(0,max(dens)),main="")
title(main="Histogram and True density")
lines(sseq,dens,col="red",lwd=2)
# }