rgnvmix function

(Quasi-)Random Number Generator for Grouped Normal Variance Mixtures

Generate vectors of random variates from grouped normal variance mixtures (including Student t with multiple degrees-of-freedom).

rgnvmix(n, qmix, groupings = 1:d, loc = rep(0, d), scale = diag(2),
       factor = NULL, method = c("PRNG", "sobol", "ghalton"), skip = 0, ...)
rgStudent(n, groupings = 1:d, df, loc = rep(0, d), scale = diag(2),
          factor = NULL, method = c("PRNG", "sobol", "ghalton"), skip = 0)

Arguments

• n: sample size $n$ (positive integer).

• qmix: specification of the mixing variables $W_i$; see pgnvmix().

• groupings: vector specifying the group structure; see pgnvmix().

• df: vector specifying the degrees-of-freedom; see see pgStudent().

• loc: see pgnvmix().

• scale: see pgnvmix(). scale must be positive definite; sampling from singular normal variance mixtures can be achieved by providing factor.

• factor: see rnvmix().

• method: see rnvmix().

• skip: see rnvmix().

• ...: additional arguments (for example, parameters) passed to the underlying mixing distribution when qmix

  is a character string or an element of qmix is a function.

Returns

rgnvmix() returns an $(n, d)$-matrix

containing $n$ samples of the specified (via qmix) $d$-dimensional grouped normal variance mixture with location vector loc and scale matrix scale

(a covariance matrix).

rgStudent() returns samples from the $d$-dimensional multivariate t distribution with multiple degrees-of-freedom specified by df, location vector loc and scale matrix scale.

Details

Internally used is factor, so scale is not required to be provided if factor is given.

The default factorization used to obtain factor is the Cholesky decomposition via chol(). To this end, scale

needs to have full rank.

rgStudent() is a wrapper of rgnvmix(, qmix = "inverse.gamma", df = df).

Author(s)

Erik Hintz, Marius Hofert and Christiane Lemieux

References

Hintz, E., Hofert, M. and Lemieux, C. (2020), Grouped Normal Variance Mixtures. Risks 8(4), 103.

Hintz, E., Hofert, M. and Lemieux, C. (2021), Normal variance mixtures: Distribution, density and parameter estimation. Computational Statistics and Data Analysis 157C, 107175.

Hintz, E., Hofert, M. and Lemieux, C. (2022), Multivariate Normal Variance Mixtures in : The Package nvmix. Journal of Statistical Software, tools:::Rd_expr_doi("10.18637/jss.v102.i02") .

McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.

See Also

rnvmix(), pgnvmix()

Examples

n <- 1000 # sample size

## Generate a random correlation matrix in d dimensions
d <- 2
set.seed(157)
A <- matrix(runif(d * d), ncol = d)
scale <- cov2cor(A %*% t(A))

## Example 1: Exponential mixture
## Let W_1 ~ Exp(1), W_2 ~ Exp(10)
rates <- c(1, 10)
#qmix <- list(list("exp", rate = rates[1]), list("exp", rate = rates[2]))
qmix <- lapply(1:2, function(i) list("exp", rate = rates[i]))
set.seed(1)
X.exp1 <- rgnvmix(n, qmix = qmix, scale = scale)
## For comparison, consider NVM distribution with W ~ Exp(1)
set.seed(1)
X.exp2 <- rnvmix(n, qmix = list("exp", rate = rates[1]), scale = scale)
## Plot both samples with the same axes
opar <- par(no.readonly = TRUE)
par(mfrow=c(1,2))
plot(X.exp1, xlim = range(X.exp1, X.exp2), ylim = range(X.exp1, X.exp2),
     xlab = expression(X[1]), ylab = expression(X[2]))
mtext("Two groups with rates 1 and 10")
plot(X.exp2, xlim = range(X.exp1, X.exp2), ylim = range(X.exp1, X.exp2),
     xlab = expression(X[1]), ylab = expression(X[2]))
mtext("One group with rate 1")
par(opar)

## Example 2: Exponential + Inverse-gamma mixture
## Let W_1 ~ Exp(1), W_2 ~ IG(1.5, 1.5) (=> X_2 ~ t_3 marginally)
df <- 3
qmix <- list(list("exp", rate = rates[1]),
             function(u, df) 1/qgamma(1-u, shape = df/2, rate = df/2))
set.seed(1)
X.mix1 <- rgnvmix(n, qmix = qmix, scale = scale, df = df)
plot(X.mix1, xlab = expression(X[1]), ylab = expression(X[2]))

## Example 3: Mixtures in d > 2
d <- 5
set.seed(157)
A <- matrix(runif(d * d), ncol = d)
scale <- cov2cor(A %*% t(A))

## Example 3.1: W_i ~ Exp(i), i = 1,...,d
qmix <- lapply(1:d, function(i) list("exp", rate = i))
set.seed(1)
X.mix2 <- rgnvmix(n, qmix = qmix, scale = scale)

## Example 3.2: W_1, W_2 ~ Exp(1), W_3, W_4, W_5 ~ Exp(2)
## => 2 groups, so we need two elements in 'qmix'
qmix <- lapply(1:2, function(i) list("exp", rate = i))
groupings <- c(1, 1, 2, 2, 2)
set.seed(1)
X.mix3 <- rgnvmix(n, qmix = qmix, groupings = groupings, scale = scale)

## Example 3.3: W_1, W_3 ~ IG(1, 1), W_2, W_4 ~ IG(2, 2), W_5 = 1
## => X_1, X_3 ~ t_2; X_2, X_4 ~ t_4, X_5 ~ N(0, 1)
qmix <- list(function(u, df1) 1/qgamma(1-u, shape = df1/2, rate = df1/2),
             function(u, df2) 1/qgamma(1-u, shape = df2/2, rate = df2/2),
             function(u) rep(1, length(u)))
groupings = c(1, 2, 1, 2, 3)
df = c(2, 4, Inf)
set.seed(1)
X.t1 <- rgnvmix(n, qmix = qmix, groupings = groupings, scale = scale,
                df1 = df[1], df2 = df[2])

## This is equivalent to calling 'rgnmvix' with 'qmix = "inverse.gamma"'
set.seed(1)
X.t2 <- rgnvmix(n, qmix = "inverse.gamma", groupings = groupings, scale = scale,
                df = df)

## Alternatively, one can use the user friendly wrapper 'rgStudent()'
set.seed(1)
X.t3 <- rgStudent(n, df = df, groupings = groupings, scale = scale)

stopifnot(all.equal(X.t1, X.t2), all.equal(X.t1, X.t3))