Density of Grouped Normal Variance Mixtures
Evaluating grouped normal variance mixture density functions (including Student t with multiple degrees-of-freedom).
dgnvmix(x, groupings = 1:d, qmix, loc = rep(0, d), scale = diag(d), factor = NULL,
factor.inv = NULL, control = list(), log = FALSE, verbose = TRUE, ...)
dgStudent(x, groupings = 1:d, df, loc = rep(0, d), scale = diag(d), factor = NULL,
factor.inv = NULL, control = list(), log = FALSE, verbose = TRUE)x |
see |
groupings |
see |
qmix |
specification of the mixing variables W_i via quantile
functions; see |
loc |
see |
scale |
see |
factor |
(d, d) lower triangular |
factor.inv |
inverse of |
df |
|
control |
|
log |
|
verbose |
see |
... |
additional arguments (for example, parameters) passed
to the underlying mixing distribution when |
Internally used is factor.inv, so factor and scale are not
required to be provided (but allowed for consistency with other functions in the
package).
dgStudent() is a wrapper of
dgnvmix(, qmix = "inverse.gamma", df = df). If there is no grouping,
the analytical formula for the density of a multivariate t distribution
is used.
Internally, an adaptive randomized Quasi-Monte Carlo (RQMC) approach
is used to estimate the log-density. It is an iterative algorithm that
evaluates the integrand at a randomized Sobol' point-set (default) in
each iteration until the pre-specified error tolerance
control$dnvmix.reltol in the control argument
is reached for the log-density. The attribute
"numiter" gives the worst case number of such iterations needed
(over all rows of x). Note that this function calls underlying
C code.
Algorithm specific parameters (such as above mentioned control$dnvmix.reltol)
can be passed as a list via the control argument,
see get_set_param() for details and defaults.
If the error tolerance cannot be achieved within control$max.iter.rqmc
iterations and fun.eval[2] function evaluations, an additional
warning is thrown if verbose=TRUE.
dgnvmix() and dgStudent() return a
numeric n-vector with the computed density values
and corresponding attributes "abs. error" and "rel. error"
(error estimates of the RQMC estimator) and "numiter" (number of iterations).
Erik Hintz, Marius Hofert and Christiane Lemieux
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.
rgnvmix(), pgnvmix(), get_set_param()
n <- 100 # sample size to generate evaluation points
### 1. Inverse-gamma mixture
## 1.1. Grouped t with mutliple dof
d <- 3 # dimension
set.seed(157)
A <- matrix(runif(d * d), ncol = d)
P <- cov2cor(A %*% t(A)) # random scale matrix
df <- c(1.1, 2.4, 4.9) # dof for margin i
groupings <- 1:d
x <- rgStudent(n, df = df, scale = P) # evaluation points for the density
### Call 'dgnvmix' with 'qmix' a string:
set.seed(12)
dgt1 <- dgnvmix(x, qmix = "inverse.gamma", df = df, scale = P)
### Version providing quantile functions of the mixing distributions as list
qmix_ <- function(u, df) 1 / qgamma(1-u, shape = df/2, rate = df/2)
qmix <- list(function(u) qmix_(u, df = df[1]), function(u) qmix_(u, df = df[2]),
function(u) qmix_(u, df = df[3]))
set.seed(12)
dgt2 <- dgnvmix(x, groupings = groupings, qmix = qmix, scale = P)
### Similar, but using ellipsis argument:
qmix <- list(function(u, df1) qmix_(u, df1), function(u, df2) qmix_(u, df2),
function(u, df3) qmix_(u, df3))
set.seed(12)
dgt3 <- dgnvmix(x, groupings = groupings, qmix = qmix, scale = P, df1 = df[1],
df2 = df[2], df3 = df[3])
### Using the wrapper 'dgStudent()'
set.seed(12)
dgt4 <- dgStudent(x, groupings = groupings, df = df, scale = P)
stopifnot(all.equal(dgt1, dgt2, dgt3, dgt4, tol = 1e-5, check.attributes = FALSE))
## 1.2 Classical multivariate t
df <- 2.4
groupings <- rep(1, d) # same df for all components
x <- rStudent(n, scale = P, df = df) # evaluation points for the density
dt1 <- dStudent(x, scale = P, df = df, log = TRUE) # uses analytical formula
## If 'qmix' provided as string and no grouping, dgnvmix() uses analytical formula
dt2 <- dgnvmix(x, qmix = "inverse.gamma", groupings = groupings, df = df, scale = P, log = TRUE)
stopifnot(all.equal(dt1, dt2))
## Provide 'qmix' as a function to force estimation in 'dgnvmix()'
dt3 <- dgnvmix(x, qmix = qmix_, groupings = groupings, df = df, scale = P, log = TRUE)
stopifnot(all.equal(dt1, dt3, tol = 5e-4, check.attributes = FALSE))
### 2. More complicated mixutre
## Let W1 ~ IG(1, 1), W2 = 1, W3 ~ Exp(1), W4 ~ Par(2, 1), W5 = W1, all comonotone
## => X1 ~ t_2; X2 ~ normal; X3 ~ Exp-mixture; X4 ~ Par-mixture; X5 ~ t_2
d <- 5
set.seed(157)
A <- matrix(runif(d * d), ncol = d)
P <- cov2cor(A %*% t(A))
b <- 3 * runif(d) * sqrt(d) # random upper limit
groupings <- c(1, 2, 3, 4, 1) # since W_5 = W_1
qmix <- list(function(u) qmix_(u, df = 2), function(u) rep(1, length(u)),
list("exp", rate=1), function(u) (1-u)^(-1/2)) # length 4 (# of groups)
x <- rgnvmix(n, groupings = groupings, qmix = qmix, scale = P)
dg <- dgnvmix(x, groupings = groupings, qmix = qmix, scale = P, log = TRUE)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.