# Quantiles of the Multivariate t Distribution Computes the equicoordinate quantile function of the multivariate t distribution for arbitrary correlation matrices based on inversion of `pmvt`, using a stochastic root finding algorithm described in Bornkamp (2018). ```r 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, seed = NULL, ...) ``` ## 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 $P[X \le x] = p$, `upper.tail` gives $x$ with $P[X > x] = p$ and `both.tails` leads to $x$ with $P[-x \le X \le x] = p$. - `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 internally. If `corr` is given, it is assumed that appropriate standardization was performed by the user. 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. The choice `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. - `seed`: an object specifying if and how the random number generator should be initialized, see `simulate`. - `...`: 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. ## Returns 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`. ## References Bornkamp, B. (2018). Calculating quantiles of noisy distribution functions using local linear regressions. **Computational Statistics**, 33 , 487--501. ## See Also `pmvnorm`, `qmvnorm` ## Examples ```r ## 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)) ```

qmvt function