# mt: The multivariate Student's _t_ distribution In mnormt: The Multivariate Normal and t Distributions, and Their Truncated Versions

## Description

The probability density function, the distribution function and random number generation for the multivariate Student's t distribution

## Usage

 ```1 2 3 4 5 6``` ```dmt(x, mean = rep(0, d), S, df=Inf, log = FALSE) pmt(x, mean = rep(0, d), S, df=Inf, ...) rmt(n = 1, mean = rep(0, d), S, df=Inf, sqrt=NULL) sadmvt(df, lower, upper, mean, S, maxpts = 2000*d, abseps = 1e-06, releps = 0) biv.nt.prob(df, lower, upper, mean, S) ptriv.nt(df, x, mean, S) ```

## Arguments

 `x` either a vector of length `d` or (for `dmt` and `pmt`) a matrix with `d` columns, where `d=ncol(S)`, giving the coordinates of the point(s) where the density must be evaluated. `mean` either a vector of length `d`, representing the location parameter (equal to the mean vector when `df>1`), or (for `dmt` and `pmt`) a matrix whose rows represent different mean vectors; in the matrix case, its dimensions must match those of `x`. `S` a symmetric positive-definite matrix representing the scale matrix of the distribution, such that `S*df/(df-2)` is the variance-covariance matrix when `df>2`; a vector of length `1` is also allowed (in this case, `d=1` is set). `df` the degrees of freedom. For `rmt`, it must be a positive real value or `Inf`. For all other functions, it must be a positive integer or `Inf`. A value `df=Inf` is translated to a call to a suitable function for the the multivariate normal distribution. See ‘Details’ for its effect for the evaluation of distribution functions and other probabilities. `log` a logical value(default value is `FALSE`); if `TRUE`, the logarithm of the density is computed. `sqrt` if not `NULL` (default value is `NULL`), a square root of the intended scale matrix `S`; see ‘Details’ for a full description. `...` arguments passed to `sadmvt`, among `maxpts`, `absrel`, `releps`. `n` the number of random vectors to be generated `lower` a numeric vector of lower integration limits of the density function; must be of maximal length `20`; `+Inf` and `-Inf` entries are allowed. `upper` a numeric vector of upper integration limits of the density function; must be of maximal length `20`; `+Inf` and `-Inf` entries are allowed `maxpts` the maximum number of function evaluations (default value: `2000*d`) `abseps` absolute error tolerance (default value: `1e-6`). `releps` relative error tolerance (default value: `0`).

## Details

The dimension `d` cannot exceed `20` for `pmt`.

The functions `sadmvt`, `ptriv.mt` and `biv.nt.prob` are interfaces to Fortran-77 routines by Alan Genz, available from his web page; they makes use of some auxiliary functions whose authors are indicated in the Fortran code itself. The routine `sadmvt` uses an adaptive integration method. If `df=3`, a call to `pmt` activates a call to `ptriv.nt` which is specific for the trivariate case, and uses Genz's Fortran code `tvpack.f`; see Genz (2004) for the background methodology. A similar fact takes place when `df=2` with function `biv.nt.prob`; note however that the underlying Fortran code is taken from `mvtdstpack.f`, not from `tvpack.f`. If `pmt` is called with `d>3`, this is converted into a suitable call to `sadmvt`.

If `sqrt=NULL` (default value), the working of `rmt` involves computation of a square root of `S` via the Cholesky decomposition. If a non-`NULL` value of `sqrt` is supplied, it is assumed that it represents a square root of the scale matrix, otherwise represented by `S`, whose value is ignored in this case. This mechanism is intended primarily for use in a sequence of calls to `rmt`, all sampling from a distribution with fixed scale matrix; a suitable matrix `sqrt` can then be computed only once beforehand, avoiding that the same operation is repeated multiple times along the sequence of calls. For examples of use of this argument, see those in the documentation of `rmnorm`. Another use of `sqrt` is to supply a different form of square root of the scale matrix, in place of the Cholesky factor.

For efficiency reasons, `rmt` does not perform checks on the supplied arguments.

## Value

`dmt` returns a vector of density values (possibly log-transformed); `pmt` and `sadmvt` return a single probability with attributes giving details on the achieved accuracy, provided `x` of `pmnorm` is a vector; `rmt` returns a matrix of `n` rows of random vectors

## Note

The attributes `error` and `status` of the probability returned by `sadmvt` and by `pmt` (the latter only if `x` is a vector and `d>2`) indicate whether the function had a normal termination, achieving the required accuracy. If this is not the case, re-run the function with a higher value of `maxpts`.

## Author(s)

Fortran code of `SADMVT` and most auxiliary functions by Alan Genz; some additional auxiliary functions by people referred to within his program; interface to R and additional R code (for `dmt`, `rmt` etc.) by Adelchi Azzalini.

## References

Genz, A.: Fortran-77 code in files `mvt.f`, `mvtdstpack.f` and codetvpack, downloaded in 2005 and again in 2007 from his webpage, whose URL as of 2020-06-01 is http://www.math.wsu.edu/faculty/genz/software/software.html

Genz, A. (2004). Numerical computation of rectangular bivariate and trivariate normal and t probabilities. Statistics and Computing 14, 251-260.

Dunnett, C.W. and Sobel, M. (1954). A bivariate generalization of Student's t-distribution with tables for certain special cases. Biometrika 41, 153–169.

`dt`, `rmnorm` for use of argument `sqrt`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```x <- seq(-2,4,length=21) y <- 2*x+10 z <- x+cos(y) mu <- c(1,12,2) Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) df <- 4 f <- dmt(cbind(x,y,z), mu, Sigma,df) p1 <- pmt(c(2,11,3), mu, Sigma, df) p2 <- pmt(c(2,11,3), mu, Sigma, df, maxpts=10000, abseps=1e-8) x <- rmt(10, mu, Sigma, df) p <- sadmvt(df, lower=c(2,11,3), upper=rep(Inf,3), mu, Sigma) # upper tail # p0 <- pmt(c(2,11), mu[1:2], Sigma[1:2,1:2], df=5) p1 <- biv.nt.prob(5, lower=rep(-Inf,2), upper=c(2, 11), mu[1:2], Sigma[1:2,1:2]) p2 <- sadmvt(5, lower=rep(-Inf,2), upper=c(2, 11), mu[1:2], Sigma[1:2,1:2]) c(p0, p1, p2, p0-p1, p0-p2) ```