Random generation for the Asymmetric Power Distribution with parameters theta, phi, alpha and lambda.
This generator is called by function gensample to create random variables based on its parameters.
If theta, phi, alpha and lambda are not specified they assume the default values of 0, 1, 0.5 and 2, respectively.
The Asymmetric Power Distribution with parameters theta,
phi, alpha and lambda has density:
$$f(u) = \frac{1}{\phi}\frac{\delta^{1/\lambda}_{\alpha,\lambda}}{\Gamma(1+1/\lambda)}\exp\left[-\frac{\delta_{\alpha,\lambda}}{\alpha^{\lambda}}\left|\frac{u-\theta}{\phi}\right|^{\lambda}\right]$$
if $$u\leq0$$ and
$$f(u) =
\frac{1}{\phi}\frac{\delta^{1/\lambda}_{\alpha,\lambda}}{\Gamma(1+1/\lambda)}\exp\left[-\frac{\delta_{\alpha,\lambda}}{(1-\alpha)^{\lambda}}\left|\frac{u-\theta}{\phi}\right|^{\lambda}\right]$$
if $$u\leq0,$$where \(0<\alpha<1, \lambda>0\) and \(\delta_{\alpha,\lambda}=\frac{2\alpha^{\lambda}(1-\alpha)^{\lambda}}{\alpha^{\lambda}+(1-\alpha)^{\lambda}}\).
The mean and variance of APD are defined respectively by $$E(U) = \theta+\phi\frac{\Gamma(2/\lambda)}{\Gamma(1/\lambda)} [1-2\alpha]\delta_{\alpha,\lambda}^{-1/\lambda}$$ and $$V(U) = \phi^2 \frac{\Gamma(3/\lambda)\Gamma(1/\lambda)[1-3\alpha+3\alpha^2]-\Gamma(2/\lambda)^2[1-2\alpha]^2}{\Gamma^2(1/\lambda)} \delta_{\alpha,\lambda}^{-2/\lambda}. $$
Pierre Lafaye de Micheaux, Viet Anh Tran (2016). PoweR: A Reproducible Research Tool to Ease Monte Carlo Power Simulation Studies for Goodness-of-fit Tests in R. Journal of Statistical Software, 69(3), 1--42. doi:10.18637/jss.v069.i03
Komunjer, I. (2007), Asymmetric Power Distribution: Theory and Applications to Risk Measurement, Journal of Applied Econometrics, 22, 891--921.
See Distributions for other standard distributions.
# NOT RUN {
res <- gensample(38,10000,law.pars=c(3,2,0.5,1))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)
# }
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