Description Usage Arguments Details Value Note Author(s) References See Also Examples
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
.
1 2 3 4 |
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 ≤ x] otherwise, P[X > x]. |
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 μ, scale parameter σ>0 and skewness parameter p in (0,1), if its probability density function (pdf) is given by
f(y|μ,σ,p)=\frac{p(1-p)}{σ}\exp {-ρ_{p}(\frac{y-μ}{σ})}
where ρ_p(.) is the so called check (or loss) function defined by
ρ_p(u)=u(p - I_{u<0})
, with I_{.} denoting the usual indicator function. This distribution is denoted by ALD(μ,σ,p) and it's p-th quantile is equal to μ.
The scale parameter sigma
must be positive and non zero. The skew parameter p
must be between zero and one (0<p
<1).
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 numerical arguments other than n
are recycled to the length of the result.
Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br>
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 | ## 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)
|
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