Random generation for the modified 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 modified Asymmetric Power Distribution with parameters theta
, phi
, theta1
and theta2
has density:
$$f(x\mid\boldsymbol{\theta})=\frac{(\delta_{\boldsymbol{\theta}}/2)^{1/\theta_2}}{\Gamma(1+1/\theta_2)}\exp\left[-\left(\frac{2(\delta_{\boldsymbol{\theta}}/2)^{1/\theta_2}}{1+sign(x)(1-2\theta_1)}|x|\right)^{\theta_2}\right]$$ where \(\boldsymbol{\theta}=(\theta_2, \theta_1)^T\) is the vector of parameters, \(\theta_2>0, 0<\theta_1<1\) and $$\delta_{\boldsymbol{\theta}}=\frac{2(\theta_1)^{\theta_2} (1-\theta_1)^{\theta_2}}{(\theta_1)^{\theta_2}+(1-\theta_1)^{\theta_2}}$$.
The mean and variance of APD are defined respectively by $$E(U) = \theta + 2 ^ {1 / \theta_2} \phi \Gamma(2 / \theta_2) (1 - 2 \theta_1) \delta ^ {-1 / \theta_2} / \Gamma(1 / \theta_2)$$ and $$V(U) = 2 ^ {2 / \theta_2} \phi ^ 2 \left(\Gamma(3 / \theta_2) \Gamma(1 / \theta_2) (1 - 3 \theta_1 + 3 \theta_1 ^ 2) - \Gamma^2(2 / \theta_2) (1 - 2 \theta_1) ^ 2\right) \delta ^ {-2 / \theta_2} / \Gamma^2(1 / \theta_2).$$
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
Desgagne, A. and Lafaye de Micheaux, P. and Leblanc, A. (2016), Test of normality based on alternate measures of skewness and kurtosis, ,
See Distributions
for other standard distributions.
# NOT RUN {
res <- gensample(39, 10000, law.pars = c(3, 2, 0.5, 1))
res$law
res$law.pars
mean(res$sample)
sd(res$sample)
# }
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