momentsALD {ald}

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 \mu.

sigma

scale parameter \sigma.

p

skewness parameter p.

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 pth 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

meanALD gives the mean, varALD gives the variance, skewALD gives the skewness, kurtALD gives the kurtosis, momentALD gives the kth central moment, i.e., E(y-\mu)^k and absALD gives the first absolute central moment denoted by E|y-\mu|.

Author(s)

Christian E. Galarza <[e-mail]> and Victor H. Lachos <[e-mail]>

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

ALD,likALD,mleALD

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)