| Title: | A Universal Non-Uniform Random Number Generator |
|---|---|
| Description: | A universal non-uniform random number generator for quite arbitrary distributions with piecewise twice differentiable densities. |
| Authors: | Josef Leydold, Carsten Botts and Wolfgang H\"ormann |
| Maintainer: | Josef Leydold <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 2.4 |
| Built: | 2024-12-19 06:52:46 UTC |
| Source: | CRAN |
Tinflex is a universal non-uniform random number generator
based on the acceptence-rejection method for all distributions that
have a piecewise twice differentiable density function.
Required input includes the log-density function of
the target distribution and its first and second derivatives.
| Package: | Tinflex |
| Type: | Package |
| Version: | 2.4 |
| Date: | 2023-03-21 |
| License: | GPL 2 or later |
Package Tinflex serves three purposes:
The installed package provides a fast routine for sampling from any distribution that has a piecewise twice differentiable density function.
It provides C routines functions that could be used in other packages (see the installed C header files).
The R source (including comments) presents all details of the general sampling method which are not entirely worked out in our paper cited in the see references below.
Algorithm Tinflex is a universal random variate generator based
on transformed density rejection which is a variant of the
acceptance-rejection method. The generator first computes
and stores hat and squeeze functions and then uses these functions
to generate variates from the distribution of interest. Since the
setup procedure is separated from the generation procedure, many
samples can be drawn from the same distribution without rerunning the
(expensive) setup.
The algorithm requires the following data about the distribution
(for further details see Tinflex.setup):
the log-density of the targent distribution;
its first derivative;
its second derivative (optionally);
a starting partition of its domain such that each subinterval contains at most one inflection point of the transformed density;
a transformation for the density (default is the logarithm transformation).
The following routines are provided.
Tinflex.setupcomputes hat and squeeze. The
table is then stored in a generator object of class
"Tinflex".
Tinflex.sampledraws a random sample from a particular generator object.
print.Tinflexprints the properties a generator
object of class "Tinflex".
plot.Tinflexplots density, hat and squeeze
functions for a given generator object of class "Tinflex".
For further details see Tinflex.setup.
There are variants of the method. The first one uses the second derivative to determine regions whre the transformed density is convex, concave, or has a single inflection points. The second variant estimates the signs on the second derivative by means of the first derivative. Thus it is easier to use at the expense of a more complex algorithm.
There are two different implementation:
Routine Tinflex.setup is implemented mainly in R and
serves (together with Tinflex:::Tinflex.sample.R) as
a reference implementation of the published algorithm.
Nevertheless, the sampling routine Tinflex.sample runs
quite fast.
Routine Tinflex.setup.C on the other hand is implemented
entirely in C. So it also allows to link to the underlying C code from
other packages.
It is very important to note that the user is responsible for the
correctness of the supplied arguments. Since the algorithm works (in theory)
for all distributions with piecewise twice differentiable density
functions, it is not possible to detect improper arguments. It is thus
recommended that the user inspect the generator object visually by
means of the plot method (see plot.Tinflex for
details).
Routine Tinflex.sample is implemented both as pure R
code (routine Tinflex.sample.R) for documenting the algorithm
as well as C code for fast performance.
Josef Leydold [email protected], Carsten Botts and Wolfgang Hörmann.
C. Botts, W. Hörmann, and J. Leydold (2013), Transformed Density Rejection with Inflection Points, Statistics and Computing 23(2), 251–260, doi:10.1007/s11222-011-9306-4. See also Research Report Series / Department of Statistics and Mathematics Nr. 110, Department of Statistics and Mathematics, WU Vienna University of Economics and Business, https://epub.wu.ac.at/id/eprint/3158.
W. Hörmann, and J. Leydold (2022), A Generalized Transformed Density Rejection Algorithm, in: Advances in Modeling and Simulation, Ch. 14, doi:10.1007/978-3-031-10193-9_14, accepted for publication.. See also Research Report Series / Department of Statistics and Mathematics Nr. 135, Department of Statistics and Mathematics, WU Vienna University of Economics and Business, https://research.wu.ac.at/de/publications/a-generalized-transformed-density-rejection-algorithm.
See Tinflex.setup for further details.
Package Runuran provides a set of many other automatic non-uniform sampling algorithms.
## Bimodal density ## f(x) = exp( -|x|^alpha + s*|x|^beta + eps*|x|^2 ) ## with alpha > beta >= 2 and s, eps > 0 alpha <- 4.2 beta <- 2.1 s <- 1 eps <- 0.1 ## Log-density and its derivatives. lpdf <- function(x) { -abs(x)^alpha + s*abs(x)^beta + eps*abs(x)^2 } dlpdf <- function(x) { (sign(x) * (-alpha*abs(x)^(alpha-1) + s*beta*abs(x)^(beta-1) + 2*eps*abs(x))) } d2lpdf <- function(x) { (-alpha*(alpha-1)*abs(x)^(alpha-2) + s*beta*(beta-1)*abs(x)^(beta-2) + 2*eps) } ## Parameter cT=0 (default): ## There are two inflection points on either side of 0. ib <- c(-Inf, 0, Inf) ## Create generator object. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=c(-Inf,0,Inf), rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=-2.5, to=2.5) ## ... and in transformed (log) scale. plot(gen, from=-2.5, to=2.5, is.trans=TRUE) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=c(-Inf,0,Inf), rho=1.1) Tinflex.sample(gen, n=10)## Bimodal density ## f(x) = exp( -|x|^alpha + s*|x|^beta + eps*|x|^2 ) ## with alpha > beta >= 2 and s, eps > 0 alpha <- 4.2 beta <- 2.1 s <- 1 eps <- 0.1 ## Log-density and its derivatives. lpdf <- function(x) { -abs(x)^alpha + s*abs(x)^beta + eps*abs(x)^2 } dlpdf <- function(x) { (sign(x) * (-alpha*abs(x)^(alpha-1) + s*beta*abs(x)^(beta-1) + 2*eps*abs(x))) } d2lpdf <- function(x) { (-alpha*(alpha-1)*abs(x)^(alpha-2) + s*beta*(beta-1)*abs(x)^(beta-2) + 2*eps) } ## Parameter cT=0 (default): ## There are two inflection points on either side of 0. ib <- c(-Inf, 0, Inf) ## Create generator object. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=c(-Inf,0,Inf), rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=-2.5, to=2.5) ## ... and in transformed (log) scale. plot(gen, from=-2.5, to=2.5, is.trans=TRUE) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=c(-Inf,0,Inf), rho=1.1) Tinflex.sample(gen, n=10)
Plotting methods for generator objects of classes
"Tinflex" and "TinflexC".
The plot shows the (transformed) density, hat and squeeze.
## S3 method for class 'Tinflex' plot(x, from, to, is.trans=FALSE, n=501, ...) ## S3 method for class 'TinflexC' plot(x, from, to, is.trans=FALSE, n=501, ...)## S3 method for class 'Tinflex' plot(x, from, to, is.trans=FALSE, n=501, ...) ## S3 method for class 'TinflexC' plot(x, from, to, is.trans=FALSE, n=501, ...)
x |
an object of class |
from, to
|
the range over which the function will be plotted. (numeric) |
is.trans |
if |
n |
the number of x values at which (transformed) PDF to evaluate. (integer) |
... |
arguments to be passed to methods, such as graphical
parameters (see
|
This is the print method for objects of class
"Tinflex" or "TinflexC".
It plots the given density function (blue) in the
domain (from,to) as well as hat function (red) and
squeeze (green) of the acceptance-rejection algorithm.
If is.trans is set to TRUE, then density function, hat
and squeeze are plotted on the transformed scale.
Notice that the latter only gives a sensible picture if parameter
cT is the same for all intervals.
Josef Leydold [email protected], Carsten Botts and Wolfgang Hörmann.
plot, plot.function.
See Tinflex.setup for examples.
Print methods for generator objects of class "Tinflex" or
"TinflexC".
## S3 method for class 'Tinflex' print(x, debug=FALSE, ...) ## S3 method for class 'TinflexC' print(x, debug=FALSE, ...)## S3 method for class 'Tinflex' print(x, debug=FALSE, ...) ## S3 method for class 'TinflexC' print(x, debug=FALSE, ...)
x |
an object of class |
debug |
enable/disable the display of detailed information about the object. (logical) |
... |
additional arguments to |
These are the print methods for objects of classes
"Tinflex" and "TinflexC".
Josef Leydold [email protected], Carsten Botts and Wolfgang Hörmann.
print.
Tinflex.setup.
Tinflex.setup.C.
See Tinflex.setup for examples.
Draw a random sample from a generator object of class
"Tinflex" or "TinflexC".
Tinflex.sample(gen, n=1) Tinflex.sample.C(gen, n=1)Tinflex.sample(gen, n=1) Tinflex.sample.C(gen, n=1)
gen |
an object of class |
n |
sample size. (integer) |
Routine Tinflex.sample.C allows objects of class
"TinflexC" only and thus is a bit faster than the same call
with routine Tinflex.sample.
Josef Leydold [email protected], Carsten Botts and Wolfgang Hörmann.
See Tinflex.setup for examples.
Create a generator object of class "Tinflex" or
"TinflexC".
Tinflex.setup(lpdf, dlpdf, d2lpdf=NULL, ib, cT=0, rho=1.1, max.intervals=1001) Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib, cT=0, rho=1.1, max.intervals=1001)Tinflex.setup(lpdf, dlpdf, d2lpdf=NULL, ib, cT=0, rho=1.1, max.intervals=1001) Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib, cT=0, rho=1.1, max.intervals=1001)
lpdf |
log-density of targent distribution. (function) |
dlpdf |
first derivative of log-density. (function) |
d2lpdf |
second derivative of log-density. (function, optional) |
ib |
break points for partition of domain of log-density. (numeric vector of length greater than 1) |
cT |
parameter for transformation |
rho |
performance parameter: requested upper bound for ratio of area below hat to area below squeeze. (numeric) |
max.intervals |
maximal numbers of intervals. (numeric) |
Algorithm Tinflex is a flexible algorithm that works (in
theory) for all distributions that have a piecewise twice
differentiable density function.
The algorithm is based on the transformed density rejection algorithm
which is a variant of the acceptance-rejection algorithm where
the density of the targent distribution is transformed by means of
some transformation .
Hat and squeeze functions of the density are then constructed by means
of tangents and secants.
The algorithm uses family
of transformations, where
Parameter is given by argument cT.
The algorithm requires the following input from the user:
the log-density of the targent distribution, lpdf;
its first derivative dlpdf;
its second derivative d2lpdf (optionally);
a starting partition ib of the domain of the target
distribution such that each subinterval contains at most one
inflection point of the transformed density;
the parameter(s) cT of the transformation either for
the entire domain or alternatively for each of the subintervals of
the partition.
The starting partition of the domain of the target distribution into non-overlapping intervals has to satisfy the following conditions:
The partition points must be given in ascending order (otherwise they are sorted anyway).
The first and last entry of this vector are the boundary
points of the domain of the distribution.
In the case when the domain of the distribution is unbounded, the
respective points are -Inf and Inf.
Within each interval of the partition, the transformed density possesses at most one inflection point (including all finite boundary points).
If a boundary point is infinite, or the density vanishes at the boundary point, then the transformed density must be concave near the corresponding boundary point and in the corresponding tail, respectively.
If the log-density lpdf has a pole or cusp at some
point , then this must be added to the starting partition
point. Moreover, it has to be counted as inflection point.
Moreover, in the corresponding intervals the transformed density
must be convex.
Argument d2lpdf is optional. If d2lpdf=NULL, then
a variant of the method is used, that determines intervals where the
transformed density is concave or convex without means of the second
derivative of the log-density.
Parameter cT is either a single numeric, that is, the same
transformation
is used for all subintervals of the domain,
or it can be set independently for each of these intervals.
In the latter case length(cT) must be equal to the number of
intervals, that is, equal to length(ib)-1.
For the choice of cT the following should be taken into
consideration:
cT=0 (the default) is most robust against numeric
underflow or overflow.
cT=-0.5 has the fastest marginal generation time.
One should always use cT=0 or cT=-0.5
for intervals that contain a point where the derivative of the
(log-) density vanishes (e.g., an extremum). For other values of
cT, the algorithm is less accurate.
For unbounded intervals or
, one has to select cT such that
.
For an interval that contains a pole at one of its boundary
points (i.e., there the density is unbounded), one has to select
cT such that and the
transformed density is convex.
If the transformed density is concave in some interval for a
particular value of cT, then it is concave for all smaller
values of cT.
Parameter rho is a performance parameter. It defines an upper
bound for ratio of the area below the hat function to the area below
the squeeze function. This parameter is an upper bound of the
rejection constant. More importantly, it provides an approximation to
the number of (time consuming) evalutions of the log-density
function lpdf.
For rho=1.01, the log-density function is evaluated once for a
sample of 300 random points. However, values of rho close to 1
also increase the table size and thus make the setup more expensive.
Parameter max.intervals defines the maximal number of
subintervals and thus the maximal table size. Putting an upper bound
on the table size prevents the algorithm from accidentally exhausting
all of the computer memory due to invalid input.
It is very unlikely that one has to increase the default value.
Routine Tinflex.setup returns an
object of class "Tinflex" that stores the random variate
generator (density, hat and squeeze functions, cumulated areas below
hat). For details see sources of the algorithm or execute
print(gen,debug=TRUE) with an object gen of class
"Tinflex".
Routine Tinflex.setup.C is equivalent to Tinflex.setup
but does all computations entirely in C. It returns an object of class
"TinflexC" which is equivalent to class "Tinflex" but
stores all data in an C structure instead of an R list.
It is very important to note that the user is responsible for the
correctness of the supplied arguments. Since the algorithm works (in theory)
for all distributions with piecewise twice differentiable density
functions, it is not possible to detect improper arguments. It is thus
recommended that the user inspect the generator object visually by
means of the plot method (see plot.Tinflex for
details).
Josef Leydold [email protected], Carsten Botts and Wolfgang Hörmann.
C. Botts, W. Hörmann, and J. Leydold (2013), Transformed Density Rejection with Inflection Points, Statistics and Computing 23(2), 251–260, doi:10.1007/s11222-011-9306-4. See also Research Report Series / Department of Statistics and Mathematics Nr. 110, Department of Statistics and Mathematics, WU Vienna University of Economics and Business, https://epub.wu.ac.at/id/eprint/3158.
W. Hörmann, and J. Leydold (2022), A Generalized Transformed Density Rejection Algorithm, in: Advances in Modeling and Simulation, Ch. 14, doi:10.1007/978-3-031-10193-9_14, accepted for publication.. See also Research Report Series / Department of Statistics and Mathematics Nr. 135, Department of Statistics and Mathematics, WU Vienna University of Economics and Business, https://research.wu.ac.at/de/publications/a-generalized-transformed-density-rejection-algorithm.
See Tinflex.sample for drawing random samples,
plot.Tinflex and print.Tinflex for
printing and plotting objects of class "Tinflex".
## Example 1: Bimodal density ## Density f(x) = exp( -|x|^alpha + s*|x|^beta + eps*|x|^2 ) ## with alpha > beta >= 2 and s, eps > 0 alpha <- 4.2 beta <- 2.1 s <- 1 eps <- 0.1 ## Log-density and its derivatives. lpdf <- function(x) { -abs(x)^alpha + s*abs(x)^beta + eps*abs(x)^2 } dlpdf <- function(x) { (sign(x) * (-alpha*abs(x)^(alpha-1) + s*beta*abs(x)^(beta-1) + 2*eps*abs(x))) } d2lpdf <- function(x) { (-alpha*(alpha-1)*abs(x)^(alpha-2) + s*beta*(beta-1)*abs(x)^(beta-2) + 2*eps) } ## Parameter cT=0 (default): ## There are two inflection points on either side of 0. ib <- c(-Inf, 0, Inf) ## Create generator object. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=c(-Inf,0,Inf), rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=-2.5, to=2.5) ## ... and in transformed (log) scale. plot(gen, from=-2.5, to=2.5, is.trans=TRUE) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=c(-Inf,0,Inf), rho=1.1) Tinflex.sample(gen, n=10) ## ------------------------------------------------------------------- ## Example 2: Exponential Power Distribution ## Density f(x) = exp( -|x|^alpha ) ## with alpha > 0 [ >= 0.015 due to limitations of FPA ] alpha <- 0.5 ## Log-density and its derivatives. lpdf <- function(x) { -abs(x)^alpha } dlpdf <- function(x) { if (x==0) {0} else {-sign(x) * alpha*abs(x)^(alpha-1)}} d2lpdf <- function(x) { -alpha*(alpha-1)*abs(x)^(alpha-2) } ## Parameter cT=-0.5: ## There are two inflection points on either side of 0 and ## a cusp at 0. Thus we need a partition point that separates ## the inflection points from the cusp. ib <- c(-Inf, -(1-alpha)/2, 0, (1-alpha)/2, Inf) ## Create generator object with c = -0.5. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=ib, cT=-0.5, rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample. Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=-4, to=4) ## ... and in transformed (log) scale. plot(gen, from=-4, to=4, is.trans=TRUE) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=ib, cT=-0.5, rho=1.1) Tinflex.sample(gen, n=10) ## ------------------------------------------------------------------- ## Example 3: Generalized Inverse Gaussian Distribution ## Density f(x) = x^(lambda-1) * exp(-omega/2 * (x+1/x)) x>= 0 ## with 0 < lambda < 1 and 0 < omega <= 0.5 la <- 0.4 ## lambda om <- 1.e-7 ## omega ## Log-density and its derivatives. lpdf <- function(x) { ifelse (x==0., -Inf, ((la - 1) * log(x) - om/2*(x+1/x))) } dlpdf <- function(x) { if (x==0) { Inf} else {(om + 2*(la-1)*x-om*x^2)/(2*x^2)} } d2lpdf <- function(x) { if (x==0) {-Inf} else {-(om - x + la*x)/x^3} } ## Parameter cT=0 near 0 and cT=-0.5 at tail: ib <- c(0, (3/2*om/(1-la) + 2/9*(1-la)/om), Inf) cT <- c(0,-0.5) ## Create generator object. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=ib, cT=cT, rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample. Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=0, to=5) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=ib, cT=cT, rho=1.1) Tinflex.sample(gen, n=10)## Example 1: Bimodal density ## Density f(x) = exp( -|x|^alpha + s*|x|^beta + eps*|x|^2 ) ## with alpha > beta >= 2 and s, eps > 0 alpha <- 4.2 beta <- 2.1 s <- 1 eps <- 0.1 ## Log-density and its derivatives. lpdf <- function(x) { -abs(x)^alpha + s*abs(x)^beta + eps*abs(x)^2 } dlpdf <- function(x) { (sign(x) * (-alpha*abs(x)^(alpha-1) + s*beta*abs(x)^(beta-1) + 2*eps*abs(x))) } d2lpdf <- function(x) { (-alpha*(alpha-1)*abs(x)^(alpha-2) + s*beta*(beta-1)*abs(x)^(beta-2) + 2*eps) } ## Parameter cT=0 (default): ## There are two inflection points on either side of 0. ib <- c(-Inf, 0, Inf) ## Create generator object. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=c(-Inf,0,Inf), rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=-2.5, to=2.5) ## ... and in transformed (log) scale. plot(gen, from=-2.5, to=2.5, is.trans=TRUE) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=c(-Inf,0,Inf), rho=1.1) Tinflex.sample(gen, n=10) ## ------------------------------------------------------------------- ## Example 2: Exponential Power Distribution ## Density f(x) = exp( -|x|^alpha ) ## with alpha > 0 [ >= 0.015 due to limitations of FPA ] alpha <- 0.5 ## Log-density and its derivatives. lpdf <- function(x) { -abs(x)^alpha } dlpdf <- function(x) { if (x==0) {0} else {-sign(x) * alpha*abs(x)^(alpha-1)}} d2lpdf <- function(x) { -alpha*(alpha-1)*abs(x)^(alpha-2) } ## Parameter cT=-0.5: ## There are two inflection points on either side of 0 and ## a cusp at 0. Thus we need a partition point that separates ## the inflection points from the cusp. ib <- c(-Inf, -(1-alpha)/2, 0, (1-alpha)/2, Inf) ## Create generator object with c = -0.5. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=ib, cT=-0.5, rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample. Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=-4, to=4) ## ... and in transformed (log) scale. plot(gen, from=-4, to=4, is.trans=TRUE) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=ib, cT=-0.5, rho=1.1) Tinflex.sample(gen, n=10) ## ------------------------------------------------------------------- ## Example 3: Generalized Inverse Gaussian Distribution ## Density f(x) = x^(lambda-1) * exp(-omega/2 * (x+1/x)) x>= 0 ## with 0 < lambda < 1 and 0 < omega <= 0.5 la <- 0.4 ## lambda om <- 1.e-7 ## omega ## Log-density and its derivatives. lpdf <- function(x) { ifelse (x==0., -Inf, ((la - 1) * log(x) - om/2*(x+1/x))) } dlpdf <- function(x) { if (x==0) { Inf} else {(om + 2*(la-1)*x-om*x^2)/(2*x^2)} } d2lpdf <- function(x) { if (x==0) {-Inf} else {-(om - x + la*x)/x^3} } ## Parameter cT=0 near 0 and cT=-0.5 at tail: ib <- c(0, (3/2*om/(1-la) + 2/9*(1-la)/om), Inf) cT <- c(0,-0.5) ## Create generator object. gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf, ib=ib, cT=cT, rho=1.1) ## Print data about generator object. print(gen) ## Draw a random sample. Tinflex.sample(gen, n=10) ## Inspect hat and squeeze visually in original scale plot(gen, from=0, to=5) ## With Version 2.0 the setup also works without providing the ## second derivative of the log-density gen <- Tinflex.setup.C(lpdf, dlpdf, d2lpdf=NULL, ib=ib, cT=cT, rho=1.1) Tinflex.sample(gen, n=10)