dgnvmix function

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)

Arguments

• x: see dnvmix().

• groupings: see pgnvmix().

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

• loc: see pnvmix().

• scale: see pnvmix(); must be positive definite.

• factor: $(d, d)$ lower triangular matrix

  such that factor %*% t(factor) equals scale. Internally used is factor.inv.

• factor.inv: inverse of factor; if not provided, computed via solve(factor).

• df: vector of length length(unique(groupings)) so that variable i has degrees-of-freedom df[groupings[i]]; all elements must be positive and can be Inf, in which case the corresponding marginal is normally distributed.

• control: list specifying algorithm specific parameters; see get_set_param().

• log: logical indicating whether the logarithmic density is to be computed.

• verbose: see pnvmix().

• ...: 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

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).

Details

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.

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

rgnvmix(), pgnvmix(), get_set_param()

Examples

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, tol = 1e-5, check.attributes = FALSE),
          all.equal(dgt1, dgt3, tol = 1e-5, check.attributes = FALSE),
          all.equal(dgt1, 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)