gammamix function

Functionalities for Gamma Scale Mixture Models

Evaluating density-, distribution- and quantile-function of Gamma scale mixtures as well as random variate generation.

dgammamix(x, qmix, d, control = list(), verbose = TRUE, log = FALSE, ...)
pgammamix(x, qmix, d, lower.tail = TRUE, control = list(), verbose = TRUE, ...)
qgammamix(u, qmix, d, control = list(), verbose = TRUE, q.only = TRUE,
          stored.values = NULL, ...)
rgammamix(n, rmix, qmix, d, method = c("PRNG", "sobol", "ghalton"),
          skip = 0, ...)

Arguments

• x: $n$-vector of evaluation points.

• u: $n$-vector of probabilities.

• qmix: see pnvmix().

• rmix: see rnvmix().

• d: dimension of the underlying normal variance mixture, see also details below.

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

• lower.tail: logical; if TRUE (default), probabilities are $P(X<= x)$, otherwise $P(X > x)$.

• log: logical indicating whether the log-density shall be returned.

• q.only: see qnvmix().

• stored.values: see qnvmix().

• method: see rnvmix().

• skip: see rnvmix().

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

• verbose: logical

  indicating whether a warning is given if the required precision has not been reached.

• ...: additional arguments (for example, parameters) passed to the underlying mixing distribution when qmix is a character string or function.

Returns

pgammamix() and dgammamix() return a numeric $n$-vector with the computed probabilities/densities and corresponding attributes "abs. error"

and "rel. error" (error estimates of the RQMC estimator) and "numiter" (number of iterations).

If q.only = TRUE, qgammamix() a vector of the same length as u with entries $q_i$ where $q_i$ satisfies $q_i = inf_x { F(x) >= u_i}$ where $F(x)$ the df of the Gamma mixture specified via qmix; if q.only = FALSE, see qnvmix.

rgammamix() returns a $n$-vector containing $n$

samples of the specified (via mix) Gamma mixture.

Details

We define a Gamma mixture as a random variable $Dsq$ satisfying, in distribution, $Dsq = W*Gamma(d/2, 2)$ where $W$ is specified via qmix. If $X$ follows a $d-$dimensional normal variance mixture, the squared Mahalanobis distance $(X-\mu)^T Sigma^{-1}(X-\mu)$ has the same distribution as $Dsq$.

The functions presented here are similar to the corresponding functions for normal variance mixtures (d/p/q/rnvmix()), details can be found in the corresponding help-files there.

Author(s)

Erik Hintz, Marius Hofert and Christiane Lemieux

References

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") .

See Also

dnvmix(), pnvmix(), qnvmix(), rnvmix(), get_set_param(), qqplot_maha(), fitnvmix()

Examples

## Specify inverse-gamma mixture => results in d * F(d, nu) dist'n,
## handled correctly when 'qmix = "inverse.gamma"' is specified
qmix <- function(u, nu) 1/qgamma(1 - u, shape = nu/2, rate = nu/2)

## Example for rgammamix()
set.seed(271) # for reproducibility
n  <- 25
nu <- 3
d  <- 5
x  <- rgammamix(n, qmix = qmix, d = d, nu = nu)

## Evaluate distribution function at 'x'
p.true_1 <- pgammamix(x, qmix = "inverse.gamma", d = d, df = nu) # calls pf(...)
p.true_2 <- pf(x/d, df1 = d, df2 = nu)
p.estim  <- pgammamix(x, qmix = qmix, d = d, nu = nu)
stopifnot(all.equal(p.true_1, p.true_2, tol = 1e-3,
                    check.attributes = FALSE),
          all.equal(p.true_1, p.estim, tol = 1e-3,
                    check.attributes = FALSE))

## Evaluate density function at 'x'
d.true_1 <- dgammamix(x, qmix = "inverse.gamma", d = d, df = nu)
d.true_2 <- df(x/d, df1 = d, df2 = nu)/d
d.est  <- dgammamix(x, qmix = qmix, d = d, nu = nu)
stopifnot(all.equal(d.true_1, d.true_2, tol = 5e-4,
                    check.attributes = FALSE),
          all.equal(d.true_1, d.est, tol = 5e-4,
                    check.attributes = FALSE))

## Evaluate quantile function
u <- seq(from = 0.5, to = 0.9, by = 0.1)
q.true_1 <- qgammamix(u, qmix = "inverse.gamma", d = d, df = nu)
q.true_2 <- qf(u, df1 = d, df2 = nu) * d
q.est  <- qgammamix(u, qmix = qmix, d = d, nu = nu)
stopifnot(all.equal(q.true_1, q.true_2, tol = 5e-4,
                    check.attributes = FALSE),
          all.equal(q.true_1, q.est, tol = 5e-4,
                    check.attributes = FALSE))