*! version 1.1 03 Jan 2019

Title

rtmvt -- Truncated Multivariate t Random Deviates

Syntax

rtmvt, [options]

Required
lowertruncation(numlist miss) Vector of lower truncation points.

uppertruncation(numlist miss) Vector of upper truncation points.

Optional
n(#) Number of random deviates to generate.

delta(numlist) Vector of non-centrality parameters.

sigma(string) Scale matrix.

df(#) Degrees of freedom.

Description

rtmvt generates random deviates from a specified truncated multivariate t distribution.
Arbitrary non-centrality parameters, scale matrices, degrees of freedom, and truncation points are supported.

It achieves the required generation using accept-reject sampling of random values generated from a corresponding multivariate t distribution, using the method of rmvt.

Options

Required

lowertruncation(numlist miss) A numlist (vector) giving the lower truncation points for the truncated multivariate t distribution under consideration.
Missing values are allowed, as . can be used to indicated a value is -Infinity.

uppertruncation(numlist miss) A numlist (vector) giving the upper truncation points for the truncated multivariate t distribution under consideration.
Its length must be equal to the length of option lowertruncation, and the elements of uppertruncation must be strictly greater than the corresponding elements in lowertruncation.
Missing values are allowed, as . can be used to indicated a value is +Infinity.

Optional

n(#) An integer giving the number of random deviates to generate from the specified truncated multivariate t distribution.
It must be greater than or equal to one.
It defaults to 1.

delta(numlist) A numlist (vector) giving the non-centrality parameters of the non-truncated multivariate t distribution from which the truncated multivariate t distribution under consideration is constructed.
If specified, its length must be equal to the length of option lowertruncation.
If left unspecified, it will default internally to the zero vector of the correct length.

sigma(string) A string (matrix) giving the scale matrix of the non-truncated multivariate t distribution from which the truncated multivariate t distribution under consideration is constructed.
If specified, it must be symmetric positive-definite, and its number of rows (equivalently columns) must be equal to the length of option lowertruncation.
If left unspecified, it will default internally to the identity matrix of the correct dimension.

df(#) A real giving the number of degrees of freedom of the non-truncated multivariate t distribution from which the truncated multivariate t distribution under consideration is constructed.
It must be either greater than or equal to zero, or missing.
Truncated multivariate normal random deviates will be computed when it is equal to zero or is missing, using the method of tmvnormalden.
It defaults to 1.

Examples

/// Example 1: Without specifying delta or sigma

rtmvt, lowertruncation(-2, -2, -2) uppertruncation(2, 2, 2) n(10)

/// Example 2: Specifying delta and sigma

mat Sigma = (1, 0.5, 0.5 \ 0.5, 1, 0.5 \ 0.5, 0.5, 1)

rtmvt, lowertruncation(-2, -2, -2) uppertruncation(2, 2, 2) n(25) delta(1, 2, 3) sigma(Sigma)

/// Example 4: Requesting truncated multivariate normal random deviates

rtmvt, lowertruncation(-2, -2, -2) uppertruncation(2, 2, 2) n(20) delta(1, 1, 1) sigma(Sigma) df(0)

Stored results

rtmvt stores the following in r():

matrices:

r(rtmvt) Generated random deviates.

Authors

Dr Michael J Grayling
Population Health Sciences Institute, Newcastle University, UK
Email: michael.grayling@newcastle.ac.uk

Prof Adrian P Mander
Centre for Trials Research, Cardiff University, Cardiff, UK

See also

References:

Grayling MJ, Mander AP (2018) Calculations involving the multivariate normal and multivariate t distributions with and without truncation. Stata J 18(4):826-43.

Kotz S, Nadarajah S (2004) Multivariate t distributions and their applications. Cambridge University Press: Cambridge, UK.

Related commands:

mvtnorm (for an overview of the functionality provided by mvtnorm)

Multivariate normal distribution

drawnorm (an official Stata command for multivariate normal random deviates)
invmvnormal (mvtnorm command for multivariate normal quantiles)
lnmvnormalden (an official Stata command for multivariate normal densities)
mvnormalcv (an official Stata command for the multivariate normal distribution function)
mvnormalden (mvtnorm command for multivariate normal densities)
pmvnormal (mvtnorm command for the multivariate normal distribution function)
rmvnormal (mvtnorm command for multivariate normal random deviates)

Multivariate t distribution

invmvt (mvtnorm command for multivariate t quantiles)
mvtden (mvtnorm command for multivariate t densities)
mvt (mvtnorm command for the multivariate t distribution function)
rmvt (mvtnorm command for multivariate t random deviates)

Truncated multivariate normal distribution

invtmvnormal (mvtnorm command for truncated multivariate normal quantiles)
rtmvnormal (mvtnorm command for truncated multivariate normal random deviates)
tmvnormal (mvtnorm command for the truncated multivariate normal distribution function)
tmvnormalden (mvtnorm command for truncated multivariate normal densities)

Truncated multivariate t distribution

invtmvt (mvtnorm command for truncated multivariate t quantiles)
rtmvt (mvtnorm command for truncated multivariate t random deviates)
tmvt (mvtnorm command for the truncated multivariate t distribution function)
tmvtden (mvtnorm command for truncated multivariate t densities)