Random generation for the Log-Pareto-tail-normal
distribution with parameters alpha, mu and sigma.
This generator is called by function gensample to create random variables based on its parameters.
If alpha, mu and sigma are not specified
they assume the default values of 1.959964, 0.0 and 1.0 respectively.
The log-Pareto-tailed normal distribution has a symmetric and continuous density that belongs to the larger family of log-regularly varying distributions (see Desgagne, 2015). This is essentially a normal density with log-Pareto tails. Using this distribution instead of the usual normal ensures whole robustness to outliers in the estimation of location and scale parameters and in the estimation of parameters of a multiple linear regression.
The density of the log-Pareto-tailed normal distribution with parameters
alpha, mu and
sigma is given by
$$g(y\mid\alpha,\mu,\sigma)=\left\{ \begin{array}{ccc} \frac{1}{\sigma}\phi\left(\frac{y-\mu}{\sigma}\right) & \textrm{ if } & \mu - \alpha\sigma \le y\le \mu + \alpha\sigma, \\ &\\ \phi(\alpha)\frac{\alpha}{|y-\mu|}\left(\frac{\log \alpha}{\log (|y-\mu|/\sigma)}\right)^\beta & \textrm{ if } & |y-\mu|\ge \alpha\sigma, \end{array} \right.$$
where \(\beta = 1+2\,\phi(\alpha)\,\alpha\log(\alpha)(1-q)^{-1}\) and \(q=\Phi(\alpha)-\Phi(-\alpha)\). The functions \(\phi(\alpha)=\frac{1}{\sqrt{2\pi}}\exp[-\frac{\alpha^2}{2}]\) and \(\Phi(\alpha)\) are respectively the p.d.f. and the c.d.f. of the standard normal distribution. The domains of the variable and the parameters are \(-\infty<y<\infty\), \(\alpha>1\), \(-\infty<\mu<\infty\) and \(\sigma>0\).
Note that the normalizing constant \(K_{(\alpha,\beta)}\) (see Desgagne, 2015, Definition 3) has been set to 1. The desirable consequence is that the core of the density, between \(\mu-\alpha\sigma\) and \(\mu+\alpha\sigma\), becomes exactly the density of the \(N(\mu,\sigma^2)\). This mass of the density corresponds to \(q\). It follows that the parameter \(\beta\) is no longer free and its value depends on \(\alpha\) as given above.
For example, if we set \(\alpha=1.959964\), we obtain \(\beta=4.083613\) and \(q=0.95\) of the mass is comprised between \(\mu-\alpha\sigma\) and \(\mu+\alpha\sigma\). Note that if one is more comfortable in choosing the central mass $q$ instead of choosing directly the parameter \(\alpha\), then it suffices to use the equation \(\alpha=\Phi^{-1}((1+q)/2)\), with the contrainst \(q>0.6826895\Leftrightarrow \alpha>1\).
The mean and variance of Log-Pareto-tail-normal are not defined.
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, Alain. Robustness to outliers in location-scale parameter model using log-regularly varying distributions. Ann. Statist. 43 (2015), no. 4, 1568--1595. doi:10.1214/15-AOS1316. http://projecteuclid.org/euclid.aos/1434546215.
See Distributions for other standard distributions.
# NOT RUN {
res <- gensample(40, 10000, law.pars = c(1.959964, 0.0, 1.0))
res$law
res$law.pars
# }
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