{smcl} {* *! version 1.3 03 Jan 2019} {vieweralsosee "" "--"}{...} {vieweralsosee "Install mvnormalden" "ssc install MVTNORM"}{...} {vieweralsosee "Help mvnormalden (if installed)" "help mvnormalden"}{...} {viewerjumpto "Syntax" "mvnormalden##syntax"}{...} {viewerjumpto "Description" "mvnormalden##description"}{...} {viewerjumpto "Options" "mvnormalden##options"}{...} {viewerjumpto "Remarks" "mvnormalden##remarks"}{...} {viewerjumpto "Examples" "mvnormalden##examples"}{...} {title:Title} {phang} {bf:mvnormalden} {hline 2} Multivariate Normal Density {marker syntax}{...} {title:Syntax} {p 8 17 2} {cmdab:mvnormalden}{cmd:,} [{it:options}] {synoptset 20 tabbed}{...} {synopthdr} {synoptline} {syntab:Required} {synopt:{opt x(numlist)}} Vector of quantiles.{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 log:density}} Print the log of the density.{p_end} {synoptline} {p2colreset}{...} {p 4 6 2} {marker description}{...} {title:Description} {pstd} {cmd:mvnormalden} evaluates the probability density function of a specified multivariate normal distribution at a given vector of quantiles. Arbitrary quantiles, means, and covariance matrices are supported. {p_end} {pstd} An official Stata command, {help lnmvnormalden}, is available that provides similar functionality to {cmd:mvnormalden}. It should in general be preferred; {cmd:mvnormalden} is provided within {bf:mvtnorm} only for completeness. {p_end} {marker options}{...} {title:Options} {dlgtab:Required} {phang} {opt x(numlist)} A numlist (vector) of quantiles at which to evaluate the probability density function. {p_end} {dlgtab:Optional} {phang} {opt me:an(numlist)} A numlist (vector) giving the mean of the multivariate normal distribution under consideration. If specified, its length must be equal to the length of option {opt x}. 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 covariance matrix of the multivariate normal distribution under consideration. If specified, It must be symmetric positive-definite, and its number of rows (equivalently columns) must be equal to the length of option {opt x}. If left unspecified, it will default internally to the identity matrix of the correct dimension. {p_end}{break}{phang} {opt log:density} If specified, this indicates that the log of the density (rather than the density itself) should be printed. {p_end} {marker examples}{...} {title:Examples} /// Example 1: Without specifying {opt me:an} or {opt s:igma} {phang} {stata mvnormalden, x(0, 0, 0)} /// Example 2: Specifying {opt me:an} and {opt s:igma} {phang} {stata mat Sigma = (1, 0.5, 0.5 \ 0.5, 1, 0.5 \ 0.5, 0.5, 1)} {phang} {stata mvnormalden, x(0, 0, 0) mean(0.5, 1, 1.5) sigma(Sigma)} /// Example 3: Requesting the log of the density with {opt log:density} {phang} {stata mat Sigma = (1, 0.25 \ 0.25, 1)} {phang} {stata mvnormalden, x(1, 1) mean(2, 2) sigma(Sigma) log} {marker results}{...} {title:Stored results} {pstd} {cmd:mvnormalden} stores the following in {cmd:r()}: {synoptset 20 tabbed}{...} {p2col 5 20 24 2: scalars:}{p_end} {synopt:{cmd:r(density)}} Computed density.{p_end} {synopt:{cmd:r(log_density)}} Log of the computed density.{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} 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} Tong YL (2012) {it:The multivariate normal distribution}. Springer-Verlag: New York, US. {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)