ald (version 1.3.1)

likALD: Log-Likelihood function for the Asymmetric Laplace Distribution

Description

Log-Likelihood function 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

likALD(y, mu = 0, sigma = 1, p = 0.5, loglik = TRUE)

Arguments

y

observation vector.

mu

location parameter \(\mu\).

sigma

scale parameter \(\sigma\).

p

skewness parameter \(p\).

loglik

logical; if TRUE (default), the Log-likelihood is return, if not just the Likelihood.

Value

likeALD returns the Log-likelihood by default and just the Likelihood if loglik = FALSE.

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. Then the Log-likelihood function is given by

$$\sum_{i=1}^{n}log(\frac{p(1-p)}{\sigma}\exp {-\rho_{p}(\frac{y_i-\mu}{\sigma})})$$.

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

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,momentsALD,mleALD

Examples

# NOT RUN {
## Let's compute the log-likelihood for a given sample

y = rALD(n=1000)
loglik = likALD(y)

#Changing the true parameters the loglik must decrease
loglik2 = likALD(y,mu=10,sigma=2,p=0.3)

loglik;loglik2
if(loglik>loglik2){print("First parameters are Better")}
# }