nvmix: rgnvmix – R documentation

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rgnvmix

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

Description

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

Usage

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`.

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).

Value

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.

Author(s)

Erik Hintz, Marius Hofert and Christiane Lemieux

References

Hintz, E., Hofert, M. and Lemieux, C. (2020) Normal variance mixtures: Distribution, density and parameter estimation. https://arxiv.org/abs/1911.03017

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

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, X.t3))

nvmix

Multivariate Normal Variance Mixtures

v0.0-5

GPL (>= 3) | file LICENCE

Authors

Marius Hofert [aut, cre], Erik Hintz [aut], Christiane Lemieux [aut]

Initial release