Description Usage Arguments Details Value Author(s) References See Also Examples
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
.
1 |
y |
observation vector. |
mu |
location parameter μ. |
sigma |
scale parameter σ. |
p |
skewness parameter p. |
loglik |
logical; if TRUE (default), the Log-likelihood is return, if not just the Likelihood. |
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. Then the Log-likelihood function is given by
∑_{i=1}^{n}log(\frac{p(1-p)}{σ}\exp {-ρ_{p}(\frac{y_i-μ}{σ})})
.
The scale parameter sigma
must be positive and non zero. The skew parameter p
must be between zero and one (0<p
<1).
likeALD
returns the Log-likelihood by default and just the Likelihood if loglik = FALSE
.
Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br>
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.
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[1] -2365.35
[1] -5769.177
[1] "First parameters are Better"
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