qrLMM (version 1.1)

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.

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

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

References

Koenker, R., Machado, J. (1999). Goodness of fit and related inference processes for quantile regression. J. Amer. Statist. Assoc. 94(3):1296-1309. Yu, K. & Moyeed, R. (2001). Bayesian quantile regression. Statistics & Probability Letters, 54(4), 437-447. 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

QRLMM

Examples

## Let's plot an Asymmetric Laplace Distribution!

##Density
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")

## 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)