momentsALD {ald} | R Documentation |
Moments for the Asymmetric Laplace Distribution
Description
Mean, variance, skewness, kurtosis, central moments w.r.t mu
and first absolute central moment for the 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
meanALD(mu=0,sigma=1,p=0.5)
varALD(mu=0,sigma=1,p=0.5)
skewALD(mu=0,sigma=1,p=0.5)
kurtALD(mu=0,sigma=1,p=0.5)
momentALD(k=1,mu=0,sigma=1,p=0.5)
absALD(sigma=1,p=0.5)
Arguments
k |
moment number. |
mu |
location parameter |
sigma |
scale parameter |
p |
skewness parameter |
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 .
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 , scale parameter
and skewness parameter
in (0,1), if its probability density function (pdf) is given by
where is the so called check (or loss) function defined by
,
with denoting the usual indicator function. This distribution is denoted by
and it's
th quantile is equal to
. The scale parameter
sigma
must be positive and non zero. The skew parameter p
must be between zero and one (0<p
<1).
Value
meanALD
gives the mean, varALD
gives the variance, skewALD
gives the skewness, kurtALD
gives the kurtosis, momentALD
gives the th central moment, i.e.,
and
absALD
gives the first absolute central moment denoted by .
Author(s)
Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br>
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
Examples
## Let's compute some moments for a Symmetric Standard Laplace Distribution.
#Third raw moment
momentALD(k=3,mu=0,sigma=1,p=0.5)
#The well known mean, variance, skewness and kurtosis
meanALD(mu=0,sigma=1,p=0.5)
varALD(mu=0,sigma=1,p=0.5)
skewALD(mu=0,sigma=1,p=0.5)
kurtALD(mu=0,sigma=1,p=0.5)
# and this guy
absALD(sigma=1,p=0.5)