Quantiles of the Multivariate t Distribution

Description

Computes the equicoordinate quantile function of the multivariate t distribution for arbitrary correlation matrices based on inversion of qmvt.

Usage

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qmvt(p, interval = NULL, tail = c("lower.tail",
     "upper.tail", "both.tails"), df = 1, delta = 0, corr = NULL,
     sigma = NULL, algorithm = GenzBretz(),
     type = c("Kshirsagar", "shifted"),
     ptol = 0.001, maxiter = 500, trace = FALSE, ...)

Arguments

p

probability.

interval

optional, a vector containing the end-points of the interval to be searched. Does not need to contain the true quantile, just used as starting values by the root-finder. If equal to NULL a guess is used.

tail

specifies which quantiles should be computed. lower.tail gives the quantile x for which P[X ≤ x] = p, upper.tail gives x with P[X > x] = p and both.tails leads to x with P[-x ≤ X ≤ x] = p.

delta

the vector of noncentrality parameters of length n, for type = "shifted" delta specifies the mode.

df

degree of freedom as integer. Normal quantiles are computed for df = 0 or df = Inf.

corr

the correlation matrix of dimension n.

sigma

the covariance matrix of dimension n. Either corr or sigma can be specified. If sigma is given, the problem is standardized. If neither corr nor sigma is given, the identity matrix in the univariate case (so corr = 1) is used for corr.

algorithm

an object of class GenzBretz or TVPACK defining the hyper parameters of this algorithm.

type

type of the noncentral multivariate t distribution to be computed. type = "Kshirsagar" corresponds to formula (1.4) in Genz and Bretz (2009) (see also Chapter 5.1 in Kotz and Nadarajah (2004)) and type = "shifted" corresponds to the formula before formula (1.4) in Genz and Bretz (2009) (see also formula (1.1) in Kotz and Nadarajah (2004)).

ptol, maxiter, trace

Parameters passed to the stochastic root-finding algorithm. Iteration stops when the 95% confidence interval for the predicted quantile is inside [p-ptol, p+ptol]. maxiter is the maximum number of iterations for the root finding algorithm. trace prints the iterations of the root finder.

...

additional parameters to be passed to GenzBretz.

Details

Only equicoordinate quantiles are computed, i.e., the quantiles in each dimension coincide. The result is seed dependend.

Value

A list with two components: quantile and f.quantile give the location of the quantile and the difference between the distribution function evaluated at the quantile and p.

See Also

pmvnorm, qmvnorm

Examples

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## basic evaluation
qmvt(0.95, df = 16, tail = "both")

## check behavior for df=0 and df=Inf
Sigma <- diag(2)
set.seed(29)
q0 <- qmvt(0.95, sigma = Sigma, df = 0,   tail = "both")$quantile
set.seed(29)
q8 <- qmvt(0.95, sigma = Sigma, df = Inf, tail = "both")$quantile
set.seed(29)
qn <- qmvnorm(0.95, sigma = Sigma, tail = "both")$quantile
stopifnot(identical(q0, q8),
          isTRUE(all.equal(q0, qn, tol = (.Machine$double.eps)^(1/3))))

## if neither sigma nor corr are provided, corr = 1 is used internally
df <- 0
set.seed(29)
qt95 <- qmvt(0.95, df = df, tail = "both")$quantile
set.seed(29)
qt95.c <- qmvt(0.95, df = df, corr  = 1, tail = "both")$quantile
set.seed(29)
qt95.s <- qmvt(0.95, df = df, sigma = 1, tail = "both")$quantile
stopifnot(identical(qt95, qt95.c),
          identical(qt95, qt95.s))

df <- 4
set.seed(29)
qt95 <- qmvt(0.95, df = df, tail = "both")$quantile
set.seed(29)
qt95.c <- qmvt(0.95, df = df, corr  = 1, tail = "both")$quantile
set.seed(29)
qt95.s <- qmvt(0.95, df = df, sigma = 1, tail = "both")$quantile
stopifnot(identical(qt95, qt95.c),
          identical(qt95, qt95.s))