{smcl}
{* *! version 1.6 03 Jan 2019}{...}
{vieweralsosee "" "--"}{...}
{vieweralsosee "Install invtmvt" "ssc install MVTNORM"}{...}
{vieweralsosee "Help invtmvt (if installed)" "help invtmvt"}{...}
{viewerjumpto "Syntax" "invtmvt##syntax"}{...}
{viewerjumpto "Description" "invtmvt##description"}{...}
{viewerjumpto "Options" "invtmvt##options"}{...}
{viewerjumpto "Remarks" "invtmvt##remarks"}{...}
{viewerjumpto "Examples" "invtmvt##examples"}{...}
{title:Title}
{phang}
{bf:invtmvt} {hline 2} Quantiles of the Truncated Multivariate t Distribution

{marker syntax}{...}
{title:Syntax}
{p 8 17 2}
{cmdab:invtmvt}{cmd:,} [{it:options}]

{synoptset 20 tabbed}{...}
{synopthdr}
{synoptline}
{syntab:Required}
{synopt:{opt p(#)}} Probability.{p_end}
{synopt:{opt lowert:runcation(numlist miss)}} Vector of lower truncation points.{p_end}
{synopt:{opt uppert:runcation(numlist miss)}} Vector of upper truncation points.{p_end}
{break}
{syntab:Optional}
{synopt:{opt me:an(numlist)}} Mean vector.{p_end}
{synopt:{opt s:igma(string)}} Covariance matrix.{p_end}
{synopt:{opt df(#)}} Degrees of freedom.{p_end}
{synopt:{opt t:ail(string)}} Which type of quantile should be computed.{p_end}
{synopt:{opt max:_iter(string)}} Maximum number of allowed iterations in the root-finding algorithm.{p_end}
{synopt:{opt tol:erance(string)}} Desired tolerance in the root-finding algorithm.{p_end}
{synopt:{opt int:egrator(string)}} Method for evaluating required multivariate normal distribution functions.{p_end}
{synopt:{opt shi:fts(#)}} Number of shifts of the Quasi-Monte Carlo integration algorithm.{p_end}
{synopt:{opt sam:ples(#)}} Number of samples in each shift of the Quasi-Monte Carlo integration algorithm.{p_end}
{synoptline}
{p2colreset}{...}
{p 4 6 2}

{marker description}{...}
{title:Description}

{pstd}
{cmd:invtmvnormal} computes equicoordinate quantiles of a specified truncated multivariate t distribution through Brent's root finding algorithm (Brent, 1973).
Arbitrary means, scale matrices, degrees of freedom, and truncation points are supported.
{p_end}

{pstd}
It evaluates requisite multivariate t distribution functions using the approach of {help mvt}.
{p_end}

{marker options}{...}
{title:Options}
{dlgtab:Required}
{phang}
{opt p(#)} A real giving the desired tail probability. It must be between zero and one inclusive.
{p_end}{break}{phang}
{opt lowert:runcation(numlist miss)} A numlist (vector) giving the lower truncation points for the truncated multivariate t distribution under consideration.
Its length must be equal to the length of option {opt low:er}, and the elements of {opt uppert:runcation} must be strictly greater than the corresponding elements in {opt lowert:runcation}.
Missing values are allowed, as . can be used to indicated a value is -Infinity.
{p_end}{break}{phang}
{opt uppert:runcation(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 {opt low:er}.
Missing values are allowed, as . can be used to indicated a value is +Infinity.

{dlgtab:Optional}
{phang}
{opt del:ta(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 {opt lowert:runcation}.
If left unspecified, it will default internally to the zero vector of the correct length.
{p_end}{break}{phang}
{opt s:igma(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 {opt lowert:runcation}.
{p_end}{break}{phang}
{opt 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 quantiles will be computed when it is equal to zero or is missing, using the method of {help invtmvnormal}.
It defaults to 1.
{p_end}{break}{phang}
{opt t:ail(string)} A string specifying which type of quantile should be computed.
It must be either lower, upper, or both.
lower gives the x such that P(X <= x) = p, upper such that P(X >= x) = p, and both such that P(-x <= X <= x) = p.
It left unspecified, it defaults internally to lower.
{p_end}{break}{phang}
{opt max:_iter(#)} An integer giving the maximum allowed number of iterations in the root-finding algorithm.
It must be greater than or equal to one.
It defaults to 1000000.
{p_end}{break}{phang}
{opt tol:erance(#)} A real giving the desired tolerance in the root-finding algorithm.
It must be strictly positive.
It defaults to 0.000001.
{p_end}{break}{phang}
{opt int:egrator(string)} Only used when {opt df} is equal to zero or is missing.
It is then a string giving the method for evaluating the required multivariate normal distribution functions.
It must be either pmvnormal (to use the method from {help pmvnormal}) or mvnormal (to use {help mvnormalcv} as released in Stata 15).
If left unspecified, it will default internally to pmvnormal.
{p_end}{break}{phang}
{opt shi:fts(#)} Not used when {opt df} is equal to zero or missing and {opt int:egrator} is equal to mvnormalcv.
It is an integer giving the number of shifts to use in the Quasi-Monte Carlo integration algorithm.
It must be strictly positive.
Theoretically, increasing {opt shi:fts} will increase run-time but reduce the error in the returned value of the integral.
It defaults to 12.
{p_end}{break}{phang}
{opt sam:ples(#)} Not used when {opt df} is equal to zero or missing and {opt int:egrator} is equal to mvnormalcv.
It is an integer giving the number of samples to use in each shift of the Quasi-Monte Carlo integration algorithm.
It must be greater than or equal to 1.
Theoretically, increasing {opt sam:ples} will increase run-time but reduce the error in the returned value of the integral.
It defaults to 1000.
{p_end}

{marker examples}{...}
{title:Examples}

/// Example 1: Without specifying {opt del:ta} or {opt s:igma}
{phang} 
{stata invtmvt, p(0.95) lowertruncation(-2, -2, -2) uppertruncation(2, 2, 2)}

/// Example 2: Specifying {opt del:ta}, {opt s:igma}, and {opt t:ail}
{phang} 
{stata mat Sigma = (1, 0.5, 0.5 \ 0.5, 1, 0.5 \ 0.5, 0.5, 1)}

{phang} 
{stata invtmvt, p(0.95) lowertruncation(-2, -2, -2) uppertruncation(2, 2, 2) delta(1, 1, 1) sigma(Sigma) tail(upper)}

/// Example 3: Requesting truncated multivariate normal quantile
{phang} 
{stata invtmvt, p(0.95) lowertruncation(-2, -2, -2) uppertruncation(2, 2, 2) delta(1, 1, 1) sigma(Sigma) df(0) tail(upper)}

{phang} 
{stata invtmvnormal, p(0.95) lowertruncation(-2, -2, -2) uppertruncation(2, 2, 2) mean(1, 1, 1) sigma(Sigma) tail(upper)}

{marker results}{...}
{title:Stored results}

{pstd}
{cmd:invtmvt} stores the following in {cmd:r()}:

{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: scalars:}{p_end}
{synopt:{cmd:r(quantile)}} Computed quantile.{p_end}
{synopt:{cmd:r(error)}} Estimated error in the computed quantile.{p_end}
{synopt:{cmd:r(flag)}} Flag indicating the progress of the root-finding algorithm. Non-zero values indicate a potential problem.{p_end}
{synopt:{cmd:r(fquantile)}} Value of the objective function at the computed quantile.{p_end}
{synopt:{cmd:r(iterations)}} Number of iterations used by the root-finding algorithm.{p_end}

{title:Authors}
{p}

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

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

{title:See also}

{bf:References:}

{phang}
Brent R (1973) {it:Algorithms for minimization without derivatives}. Prentice-Hall: New Jersey, US.

{phang}
Genz A, Bretz F (2009) {it:Computation of multivariate normal and t probabilities}. Lecture Notes in Statistics, Vol 195. Springer-Verlag: Heidelberg, Germany.

{phang}
Gibson GJ, Glasbey CA, Elston DA (1994) Monte Carlo evaluation of multivariate normal integrals and sensitivity to variate ordering.
In {it:Advances in numerical methods and applications}, ed Dimov IT, Sendov B, Vassilevski PS, 120-6. River Edge: World Scientific Publishing.

{phang}
Grayling MJ, Mander AP (2018) {browse "https://www.stata-journal.com/article.html?article=st0542":Calculations involving the multivariate normal and multivariate t distributions with and without truncation}. {it:Stata J} {bf:18}(4){bf::}826-43.

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

{bf:Related commands:}

{help mvtnorm}       (for an overview of the functionality provided by {bf:mvtnorm})

Multivariate normal distribution

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

Multivariate t distribution

{help invmvt}        ({bf:mvtnorm} command for multivariate t quantiles)
{help mvtden}        ({bf:mvtnorm} command for multivariate t densities)
{help mvt}           ({bf:mvtnorm} command for the multivariate t distribution function)
{help rmvt}          ({bf:mvtnorm} command for multivariate t random deviates)

Truncated multivariate normal distribution

{help invtmvnormal}  ({bf:mvtnorm} command for truncated multivariate normal quantiles)
{help rtmvnormal}    ({bf:mvtnorm} command for truncated multivariate normal random deviates)
{help tmvnormal}     ({bf:mvtnorm} command for the truncated multivariate normal distribution function)
{help tmvnormalden}  ({bf:mvtnorm} command for truncated multivariate normal densities)

Truncated multivariate t distribution

{help invtmvt}       ({bf:mvtnorm} command for truncated multivariate t quantiles)
{help rtmvt}         ({bf:mvtnorm} command for truncated multivariate t random deviates)
{help tmvt}          ({bf:mvtnorm} command for the truncated multivariate t distribution function)
{help tmvtden}       ({bf:mvtnorm} command for truncated multivariate t densities)