law0040.Log-Pareto-tail-normal {PoweR} | R Documentation |
The Log-Pareto-tail-normal Distribution
Description
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.
Details
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.
Author(s)
P. Lafaye de Micheaux
References
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 Also
See Distributions
for other standard distributions.
Examples
res <- gensample(40, 10000, law.pars = c(1.959964, 0.0, 1.0))
res$law
res$law.pars