mig (version 1.0)

dmig: Multivariate inverse Gaussian distribution

Description

The density of the MIG model is $$f(\boldsymbol{x}+\boldsymbol{a}) =(2\pi)^{-d/2}\boldsymbol{\beta}^{\top}\boldsymbol{\xi}|\boldsymbol{\Omega}|^{-1/2}(\boldsymbol{\beta}^{\top}\boldsymbol{x})^{-(1+d/2)}\exp\left\{-\frac{(\boldsymbol{x}-\boldsymbol{\xi})^{\top}\boldsymbol{\Omega}^{-1}(\boldsymbol{x}-\boldsymbol{\xi})}{2\boldsymbol{\beta}^{\top}\boldsymbol{x}}\right\}$$ for points in the d-dimensional half-space \(\{\boldsymbol{x} \in \mathbb{R}^d: \boldsymbol{\beta}^{\top}(\boldsymbol{x}-\boldsymbol{a}) \geq 0\}\)

Usage

dmig(x, xi, Omega, beta, shift, log = FALSE)
rmig(n, xi, Omega, beta, shift, method = c("invsim", "bm"), timeinc = 0.001)
pmig(q, xi, Omega, beta, log = FALSE, method = c("sov", "mc"), B = 10000L)

Value

for dmig, the (log)-density

for rmig, an n vector if d=1 (univariate) or an n by d matrix if d > 1

an n vector of (log) probabilities

Arguments

x

n by d matrix of quantiles

xi

d vector of location parameters \(\boldsymbol{\xi}\), giving the expected value

Omega

d by d positive definite scale matrix \(\boldsymbol{\Omega}\)

beta

d vector \(\boldsymbol{\beta}\) defining the half-space through \(\boldsymbol{\beta}^{\top}\boldsymbol{\xi}>0\)

shift

d translation for the half-space \(\boldsymbol{a}\)

log

logical; if TRUE, returns log probabilities

n

number of observations

method

string; one of inverse system (invsim, default), Brownian motion (bm)

timeinc

time increment for multivariate simulation algorithm based on the hitting time of Brownian motion, default to 1e-3.

q

n by d matrix of quantiles

B

number of Monte Carlo replications for the SOV estimator

Author

Frederic Ouimet (bm), Leo Belzile (invsim)

Leo Belzile

Details

Observations are generated using the representation as the first hitting time of a hyperplane of a correlated Brownian motion.

Examples

# Density evaluation
x <- rbind(c(1, 2), c(2,3), c(0,-1))
beta <- c(1, 0)
xi <- c(1, 1)
Omega <- matrix(c(2, -1, -1, 2), nrow = 2, ncol = 2)
dmig(x, xi = xi, Omega = Omega, beta = beta)
# Random number generation
d <- 5L
beta <- runif(d)
xi <- rexp(d)
Omega <- matrix(0.5, d, d) + diag(d)
samp <- rmig(n = 1000, beta = beta, xi = xi, Omega = Omega)
mle <- fit_mig(samp, beta = beta, method = "mle")
set.seed(1234)
d <- 2L
beta <- runif(d)
Omega <- rWishart(n = 1, df = 2*d, Sigma = matrix(0.5, d, d) + diag(d))[,,1]
xi <- rexp(d)
q <- mig::rmig(n = 10, beta = beta, Omega = Omega, xi = xi)
pmig(q, xi = xi, beta = beta, Omega = Omega)