Title:	Multivariate Product Distributions for Elliptically Contoured Distributions
Version:	0.0.5
Description:	Estimates multivariate subgaussian stable densities and probabilities as well as generates random variates using product distribution theory. A function for estimating the parameters from data to fit a distribution to data is also provided, using the method from Nolan (2013) <doi:10.1007/s00180-013-0396-7>.
Imports:	matrixStats, stabledist, libstable4u, mvtnorm, stats, cubature, Matrix
Depends:	R (≥ 3.4.0)
License:	MIT + file LICENSE
Encoding:	UTF-8
RoxygenNote:	7.2.1
Suggests:	scales, VGAMextra, expint, invgamma, LNPar, rgl, uniformly, rmarkdown, knitr, testthat (≥ 3.0.0)
Config/testthat/edition:	3
URL:	https://github.com/swihart/mvpd, https://swihart.github.io/mvpd/
BugReports:	https://github.com/swihart/mvpd/issues
NeedsCompilation:	no
Packaged:	2025-06-18 17:32:43 UTC; bruce
Author:	Bruce Swihart [aut, cre]
Maintainer:	Bruce Swihart <bruce.swihart@gmail.com>
Repository:	CRAN
Date/Publication:	2025-06-18 18:00:02 UTC

Multivariate Product Distributions

Description

The purpose of this package is to offer density, probability, and random variate generating (abbreviated as [d/p/r], respectively) functions for multivariate distributions that can be represented as a product distribution. Specifically, the package will primarily focus on the product of a multivariate normal distribution and a univariate random variable. These product distributions are called Scale Mixtures of Multivariate Normal Distributions, and for particular choices of the univariate random variable distribution the resultant product distribution may be a family of interest. For instance, the square-root of a positive stable random variable multiplied by a multivariate normal distribution is the multivariate subgaussian stable distribution. Product distribution theory is applied for implementing their computation.

Multivariate subgaussian stable distributions

dmvss – multivariate subgaussian stable distribution density

pmvss – multivariate subgaussian stable distribution probabilities

rmvss – multivariate subgaussian stable distribution random variates

pmvss_mc – Monte Carlo version of probabilities, using rmvss

fit_mvss – Fit a multivariate subgaussian stable distribution (e.g. estimate parameters given data)

Author(s)

Maintainer: Bruce Swihart bruce.swihart@gmail.com (ORCID)

See Also

Useful links:

• https://github.com/swihart/mvpd

• https://swihart.github.io/mvpd/

• Report bugs at https://github.com/swihart/mvpd/issues

Adaptive multivariate integration over hypercubes (admitting infinite limits)

Description

The function performs adaptive multidimensional integration (cubature) of (possibly) vector-valued integrands over hypercubes. It is a wrapper for cubature:::adaptIntegrate, transforming (-)Inf appropriately as described in cubature's help page (http://ab-initio.mit.edu/wiki/index.php/Cubature#Infinite_intervals).

Usage

adaptIntegrate_inf_limPD(
  f,
  lowerLimit,
  upperLimit,
  ...,
  tol.ai = 1e-05,
  fDim.ai = 1,
  maxEval.ai = 0,
  absError.ai = 0,
  doChecking.ai = FALSE
)

Arguments

Examples

## integrate Cauchy Density from -Inf to Inf
adaptIntegrate_inf_limPD(function(x) 1/pi * 1/(1+x^2), -Inf, Inf)
adaptIntegrate_inf_limPD(function(x, scale) 1/(pi*scale) * 1/(1+(x/scale)^2), -Inf, Inf, scale=4)
## integrate Cauchy Density from -Inf to -3
adaptIntegrate_inf_limPD(function(x) 1/pi * 1/(1+x^2), -Inf, -3)$int
stats::pcauchy(-3)
adaptIntegrate_inf_limPD(function(x, scale) 1/(pi*scale) * 1/(1+(x/scale)^2), -Inf, -3, scale=4)$int
stats::pcauchy(-3, scale=4)

Density for the Kolmogorov Distribution

Description

Density for the Kolmogorov Distribution

Usage

dkolm(x, nterms = 500, rep = "K3", K3cutpt = 2)

Arguments

Value

the value of the density at specified x

Examples

## see https://swihart.github.io/mvpd/articles/deep_dive_kolm.html
dkolm(1)

Multivariate Subgaussian Stable Density

Description

Computes the the density function of the multivariate subgaussian stable distribution for arbitrary alpha, shape matrices, and location vectors. See Nolan (2013).

Usage

dmvss(
  x,
  alpha = 1,
  Q = NULL,
  delta = rep(0, d),
  outermost.int = c("stats::integrate", "cubature::adaptIntegrate")[1],
  spherical = FALSE,
  subdivisions.si = 100L,
  rel.tol.si = .Machine$double.eps^0.25,
  abs.tol.si = rel.tol.si,
  stop.on.error.si = TRUE,
  tol.ai = 1e-05,
  fDim.ai = 1,
  maxEval.ai = 0,
  absError.ai = 0,
  doChecking.ai = FALSE,
  which.stable = c("libstable4u", "stabledist")[1]
)

Arguments

Value

The object returned depends on what is selected for outermost.int. In the case of the default, stats::integrate, the value is a list of class "integrate" with components:

Note: The reported abs.error is likely an under-estimate as integrate assumes the integrand was without error, which is not the case in this application.

References

Nolan, John P. "Multivariate elliptically contoured stable distributions: theory and estimation." Computational Statistics 28.5 (2013): 2067-2089.

Examples

## print("mvsubgaussPD (d=2, alpha=1.71):")
Q <- matrix(c(10,7.5,7.5,10),2)
mvpd::dmvss(x=c(0,1), alpha=1.71, Q=Q)

## more accuracy = longer runtime
mvpd::dmvss(x=c(0,1),alpha=1.71, Q=Q, abs.tol=1e-8)

Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
## print("mvsubgausPD (d=3, alpha=1.71):")
mvpd::dmvss(x=c(0,1,2), alpha=1.71, Q=Q)
mvpd::dmvss(x=c(0,1,2), alpha=1.71, Q=Q, spherical=TRUE)

## How `delta` works: same as centering
X <- c(1,1,1)
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
D <- c(0.75, 0.65, -0.35)
mvpd::dmvss(X-D, alpha=1.71, Q=Q)
mvpd::dmvss(X  , alpha=1.71, Q=Q, delta=D)

Multivariate Subgaussian Stable Density for matrix inputs

Description

Computes the the density function of the multivariate subgaussian stable distribution for arbitrary alpha, shape matrices, and location vectors. See Nolan (2013).

Usage

dmvss_mat(
  x,
  alpha = 1,
  Q = NULL,
  delta = rep(0, d),
  outermost.int = c("stats::integrate", "cubature::adaptIntegrate",
    "cubature::hcubature")[2],
  spherical = FALSE,
  subdivisions.si = 100L,
  rel.tol.si = .Machine$double.eps^0.25,
  abs.tol.si = rel.tol.si,
  stop.on.error.si = TRUE,
  tol.ai = 1e-05,
  fDim.ai = NULL,
  maxEval.ai = 0,
  absError.ai = 0,
  doChecking.ai = FALSE,
  which.stable = c("libstable4u", "stabledist")[1]
)

Arguments

Value

The object returned depends on what is selected for outermost.int. In the case of the default, stats::integrate, the value is a list of class "integrate" with components:

Note: The reported abs.error is likely an under-estimate as integrate assumes the integrand was without error, which is not the case in this application.

References

Nolan, John P. "Multivariate elliptically contoured stable distributions: theory and estimation." Computational Statistics 28.5 (2013): 2067-2089.

Examples

## print("mvsubgaussPD (d=2, alpha=1.71):")
Q <- matrix(c(10,7.5,7.5,10),2)
mvpd::dmvss(x=c(0,1), alpha=1.71, Q=Q)

## more accuracy = longer runtime
mvpd::dmvss(x=c(0,1),alpha=1.71, Q=Q, abs.tol=1e-8)

Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
## print("mvsubgausPD (d=3, alpha=1.71):")
mvpd::dmvss(x=c(0,1,2), alpha=1.71, Q=Q)
mvpd::dmvss(x=c(0,1,2), alpha=1.71, Q=Q, spherical=TRUE)

## How `delta` works: same as centering
X <- c(1,1,1)
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
D <- c(0.75, 0.65, -0.35)
mvpd::dmvss(X-D, alpha=1.71, Q=Q)
mvpd::dmvss(X  , alpha=1.71, Q=Q, delta=D)

Multivariate t-Distribution Density for matrix inputs

Description

Computes the the density function of the multivariate subgaussian stable distribution for arbitrary degrees of freedom, shape matrices, and location vectors. See Swihart and Nolan (2022).

Usage

dmvt_mat(
  x,
  df = 1,
  Q = NULL,
  delta = rep(0, d),
  outermost.int = c("stats::integrate", "cubature::adaptIntegrate",
    "cubature::hcubature")[2],
  spherical = FALSE,
  subdivisions.si = 100L,
  rel.tol.si = .Machine$double.eps^0.25,
  abs.tol.si = rel.tol.si,
  stop.on.error.si = TRUE,
  tol.ai = 1e-05,
  fDim.ai = NULL,
  maxEval.ai = 0,
  absError.ai = 0,
  doChecking.ai = FALSE
)

Arguments

Value

The object returned depends on what is selected for outermost.int. In the case of the default, stats::integrate, the value is a list of class "integrate" with components:

Note: The reported abs.error is likely an under-estimate as integrate assumes the integrand was without error, which is not the case in this application.

References

Swihart & Nolan, "The R Journal: Multivariate Subgaussian Stable Distributions in R", The R Journal, 2022

Examples

x <- c(1.23, 4.56)
mu <- 1:2
Sigma <- matrix(c(4, 2, 2, 3), ncol=2)
df01 <- mvtnorm::dmvt(x, delta = mu, sigma = Sigma, df =  1, log=FALSE) # default log = TRUE!
df10 <- mvtnorm::dmvt(x, delta = mu, sigma = Sigma, df = 10, log=FALSE) # default log = TRUE!
df30 <- mvtnorm::dmvt(x, delta = mu, sigma = Sigma, df = 30, log=FALSE) # default log = TRUE!
df01
df10
df30


dmvt_mat(
  matrix(x,ncol=2),
  df = 1,
  Q = Sigma,
  delta=mu)$int


dmvt_mat(
  matrix(x,ncol=2),
  df = 10,
  Q = Sigma,
  delta=mu)$int


dmvt_mat(
  matrix(x,ncol=2),
  df = 30,
  Q = Sigma,
  delta=mu)$int

## Q: can we do non-integer degrees of freedom?
## A: yes for both mvpd::dmvt_mat and mvtnorm::dmvt

df1.5 <- mvtnorm::dmvt(x, delta = mu, sigma = Sigma, df =  1.5, log=FALSE) # default log = TRUE!
df1.5

dmvt_mat(
  matrix(x,ncol=2),
  df = 1.5,
  Q = Sigma,
  delta=mu)$int


## Q: can we do <1 degrees of freedom but >0?
## A: yes for both mvpd::dmvt_mat and mvtnorm::dmvt

df0.5 <- mvtnorm::dmvt(x, delta = mu, sigma = Sigma, df =  0.5, log=FALSE) # default log = TRUE!
df0.5

dmvt_mat(
  matrix(x,ncol=2),
  df = 0.5,
  Q = Sigma,
  delta=mu)$int

df0.0001 <- mvtnorm::dmvt(x, delta = mu, sigma = Sigma, df =  0.0001, 
                          log=FALSE) # default log = TRUE!
df0.0001

dmvt_mat(
  matrix(x,ncol=2),
  df = 0.0001,
  Q = Sigma,
  delta=mu)$int



## Q: can we do ==0 degrees of freedom?
## A: No for both mvpd::dmvt_mat and mvtnorm::dmvt

## this just becomes normal, as per the manual for mvtnorm::dmvt
df0.0 <- mvtnorm::dmvt(x, delta = mu, sigma = Sigma, df =  0, log=FALSE) # default log = TRUE!
df0.0

## Not run: 
dmvt_mat(
  matrix(x,ncol=2),
  df = 0,
  Q = Sigma,
  delta=mu)$int 

## End(Not run)

Fit a Multivariate Subgaussian Distribution

Description

Estimates the parameters (namely, alpha, shape matrix Q, and location vector) of the multivariate subgaussian distribution for an input matrix X.

Usage

fit_mvss(x)

Arguments

Details

Using the protocols outlined in Nolan (2013), this function uses libstable4u's univariate fit functions for each component.

Value

A list with parameters from the column-wise univariate fits and the multivariate alpha and shape matrix estimates (the univ_deltas are the mult_deltas):

• univ_alphas - the alphas from the column-wise univariate fits

• univ_betas - the betas from the column-wise univariate fits

• univ_gammas - the gammas from the column-wise univariate fits

• univ_deltas - the deltas from the column-wise univariate fits

• mult_alpha - the mean(univ_alphas); equivalently the multivariate alpha estimate

• mult_Q_raw - the multivariate shape matrix estimate (before applying nearPD())

• mult_Q_posdef - the nearest positive definite multivariate shape matrix estimate, nearPD(mult_Q_raw)

References

Nolan JP (2013), Multivariate elliptically contoured stable distributions: theory and estimation. Comput Stat (2013) 28:2067–2089 DOI 10.1007/s00180-013-0396-7

See Also

Rfast::mvnorm.mle, alphastable::mfitstab.elliptical

Examples

## create a 4x4 shape matrix symMat
S <- matrix(rnorm(4*4, mean=2, sd=4),4); 
symMat <- as.matrix(Matrix::nearPD(0.5 * (S + t(S)))$mat)
symMat
## generate 10,000 r.v.'s from 4-dimensional mvss
X <- mvpd::rmvss(1e4, alpha=1.5, Q=symMat, delta=c(1,2,3,4))
## use fit_mvss to recover the parameters, compare to symMat
fmv <- mvpd::fit_mvss(X)
fmv
symMat
## then use the fitted parameters to calculate a probability:
mvpd::pmvss(lower=rep(0,4),
            upper=rep(5,4),
            alpha=fmv$mult_alpha,
            Q=fmv$mult_Q_posdef,
            delta=fmv$univ_deltas,
            maxpts.pmvnorm = 25000*10)

Multivariate Elliptically Contoured Logistic Distribution

Description

Computes the probabilities for the multivariate elliptically contoured distribution for arbitrary limits, shape matrices, and location vectors.

Usage

pmvlogis(
  lower = rep(-Inf, d),
  upper = rep(Inf, d),
  nterms = 500,
  Q = NULL,
  delta = rep(0, d),
  maxpts.pmvnorm = 25000,
  abseps.pmvnorm = 0.001,
  outermost.int = c("stats::integrate", "cubature::adaptIntegrate")[1],
  subdivisions.si = 100L,
  rel.tol.si = .Machine$double.eps^0.25,
  abs.tol.si = rel.tol.si,
  stop.on.error.si = TRUE,
  tol.ai = 1e-05,
  fDim.ai = 1,
  maxEval.ai = 0,
  absError.ai = 0,
  doChecking.ai = FALSE
)

Arguments

Value

The object returned depends on what is selected for outermost.int. In the case of the default, stats::integrate, the value is a list of class "integrate" with components:

Note: The reported abs.error is likely an under-estimate as integrate assumes the integrand was without error, which is not the case in this application.

References

Nolan JP (2013), Multivariate elliptically contoured stable distributions: theory and estimation. Comput Stat (2013) 28:2067–2089 DOI 10.1007/s00180-013-0396-7

Examples

## bivariate
U <- c(1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,10),2)
mvpd::pmvlogis(L, U, nterms=1000, Q=Q)

## trivariate
U <- c(1,1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
mvpd::pmvlogis(L, U, nterms=1000, Q=Q)

## How `delta` works: same as centering
U <- c(1,1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
D <- c(0.75, 0.65, -0.35)
mvpd::pmvlogis(L-D, U-D, nterms=100, Q=Q)
mvpd::pmvlogis(L  , U  , nterms=100, Q=Q, delta=D)

## recover univariate from trivariate
crit_val <- -1.3
Q <- matrix(c(10,7.5,7.5,7.5,20,7.5,7.5,7.5,30),3) / 10
Q
pmvlogis(c(-Inf,-Inf,-Inf), 
         c( Inf, Inf, crit_val),
         Q=Q)
plogis(crit_val, scale=sqrt(Q[3,3]))

pmvlogis(c(-Inf,     -Inf,-Inf), 
         c( Inf, crit_val, Inf ),
         Q=Q)
plogis(crit_val, scale=sqrt(Q[2,2]))

pmvlogis(c(     -Inf, -Inf,-Inf), 
         c( crit_val,  Inf, Inf ),
         Q=Q)
plogis(crit_val, scale=sqrt(Q[1,1]))
 

Multivariate Subgaussian Stable Distribution

Description

Computes the probabilities for the multivariate subgaussian stable distribution for arbitrary limits, alpha, shape matrices, and location vectors. See Nolan (2013).

Usage

pmvss(
  lower = rep(-Inf, d),
  upper = rep(Inf, d),
  alpha = 1,
  Q = NULL,
  delta = rep(0, d),
  maxpts.pmvnorm = 25000,
  abseps.pmvnorm = 0.001,
  outermost.int = c("stats::integrate", "cubature::adaptIntegrate")[1],
  subdivisions.si = 100L,
  rel.tol.si = .Machine$double.eps^0.25,
  abs.tol.si = rel.tol.si,
  stop.on.error.si = TRUE,
  tol.ai = 1e-05,
  fDim.ai = 1,
  maxEval.ai = 0,
  absError.ai = 0,
  doChecking.ai = FALSE,
  which.stable = c("libstable4u", "stabledist")[1]
)

Arguments

Value

The object returned depends on what is selected for outermost.int. In the case of the default, stats::integrate, the value is a list of class "integrate" with components:

Note: The reported abs.error is likely an under-estimate as integrate assumes the integrand was without error, which is not the case in this application.

References

Nolan JP (2013), Multivariate elliptically contoured stable distributions: theory and estimation. Comput Stat (2013) 28:2067–2089 DOI 10.1007/s00180-013-0396-7

Examples

## bivariate
U <- c(1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,10),2)
mvpd::pmvss(L, U, alpha=1.71, Q=Q)

## trivariate
U <- c(1,1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
mvpd::pmvss(L, U, alpha=1.71, Q=Q)

## How `delta` works: same as centering
U <- c(1,1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
D <- c(0.75, 0.65, -0.35)
mvpd::pmvss(L-D, U-D, alpha=1.71, Q=Q)
mvpd::pmvss(L  , U  , alpha=1.71, Q=Q, delta=D)

Monte Carlo Multivariate Subgaussian Stable Distribution

Description

Computes probabilities of the multivariate subgaussian stable distribution for arbitrary limits, alpha, shape matrices, and location vectors via Monte Carlo (thus the suffix ⁠_mc⁠).

Usage

pmvss_mc(
  lower = rep(-Inf, d),
  upper = rep(Inf, d),
  alpha = 1,
  Q = NULL,
  delta = rep(0, d),
  which.stable = c("libstable4u", "stabledist")[1],
  n = NULL
)

Arguments

Value

a number between 0 and 1, the estimated probability via Monte Carlo

References

Nolan JP (2013), Multivariate elliptically contoured stable distributions: theory and estimation. Comput Stat (2013) 28:2067–2089 DOI 10.1007/s00180-013-0396-7

Examples

## print("mvpd (d=2, alpha=1.71):")
U <- c(1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,10),2)
mvpd::pmvss_mc(L, U, alpha=1.71, Q=Q, n=1e3)
mvpd::pmvss   (L, U, alpha=1.71, Q=Q)

## more accuracy = longer runtime
mvpd::pmvss_mc(L, U, alpha=1.71, Q=Q, n=1e4)

U <- c(1,1,1)
L <- -U
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
## print("mvpd: (d=3, alpha=1.71):")
mvpd::pmvss_mc(L, U, alpha=1.71, Q=Q, n=1e3)

Random Variates for the Kolmogorov Distribution

Description

Random Variates for the Kolmogorov Distribution

Usage

rkolm(n, nterms = 500)

Arguments

Value

n random variates

Examples

## see https://swihart.github.io/mvpd/articles/deep_dive_kolm.html
rkolm(10)

Multivariate Logistic Random Variables

Description

Computes random vectors of the multivariate symmetric logistic distribution for arbitrary correlation matrices using the asymptotic Kolmogorov distribution – see references.

Usage

rmvlogis(n, Q = NULL, delta = rep(0, d), BIG = 500)

Arguments

References

Scale Mixtures of Normal Distributions Author(s): D. F. Andrews and C. L. Mallows Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 36, No. 1 (1974), pp. 99-102 Published by: Wiley for the Royal Statistical Society Stable URL: http://www.jstor.org/stable/2984774

Examples

rmvlogis(10, Q=diag(5))

## Not run: 
QMAT <- matrix(c(1,0,0,1),nrow=2)
draw_NNMD  <- NonNorMvtDist::rmvlogis(2e3, parm1=rep(0,2), parm2=rep(1,2))
draw_mvpd  <-          mvpd::rmvlogis(2e3,     Q=QMAT)

mean(draw_NNMD[,1]   < -1 & draw_NNMD[,2]   < 3)
mean(draw_mvpd[,1] < -1 & draw_mvpd[,2] < 3)

plogis(-1)
mean(draw_NNMD[,1] < -1)
mean(draw_mvpd[,1] < -1)

plogis(3)
mean(draw_NNMD[,2] < 3)
mean(draw_mvpd[,2] < 3)
 
rangex <- range(c(draw_mvpd[,1],draw_NNMD[,1]))
rangey <- range(c(draw_mvpd[,2],draw_NNMD[,2]))

par(mfrow=c(3,2), pty="s", mai=c(.5,.1,.1,.1))
plot(draw_NNMD, xlim=rangex, ylim=rangey); abline(h=0,v=0)
plot(draw_mvpd   , xlim=rangex, ylim=rangey); abline(h=0,v=0)

hist(draw_NNMD[,1]  , breaks = 100,  xlim=rangex, probability=TRUE, ylim=c(0,.40))
curve(dlogis(x), add=TRUE, col="blue",lwd=2)
hist(draw_mvpd[,1], breaks = 100,  xlim=rangex, probability=TRUE, ylim=c(0,.40))
curve(dlogis(x), add=TRUE, col="blue",lwd=2)
hist(draw_NNMD[,2]  , breaks = 100,  xlim=rangex, probability=TRUE, ylim=c(0,.40))
curve(dlogis(x), add=TRUE, col="blue",lwd=2)
hist(draw_mvpd[,2], breaks = 100,  xlim=rangex, probability=TRUE, ylim=c(0,.40))
curve(dlogis(x), add=TRUE, col="blue",lwd=2)

## End(Not run)

Multivariate Subgaussian Stable Random Variates

Description

Computes random vectors of the multivariate subgaussian stable distribution for arbitrary alpha, shape matrices, and location vectors. See Nolan (2013).

Usage

rmvss(
  n,
  alpha = 1,
  Q = NULL,
  delta = rep(0, d),
  which.stable = c("libstable4u", "stabledist")[1]
)

Arguments

Value

Returns the n by d matrix containing multivariate subgaussian stable random variates where d=nrow(Q).

References

Nolan JP (2013), Multivariate elliptically contoured stable distributions: theory and estimation. Comput Stat (2013) 28:2067–2089 DOI 10.1007/s00180-013-0396-7

Examples

## generate 10 random variates of a bivariate mvss
rmvss(n=10, alpha=1.71, Q=matrix(c(10,7.5,7.5,10),2))

## generate 10 random variates of a trivariate mvss
Q <- matrix(c(10,7.5,7.5,7.5,10,7.5,7.5,7.5,10),3)
rmvss(n=10, alpha=1.71, Q=Q)