# Density of Grouped Normal Variance Mixtures Evaluating grouped normal variance mixture density functions (including Student **t** with multiple degrees-of-freedom). ```r 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 ```r 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) ```

dgnvmix function