The Monte Carlo EM method for estimating multinomial probit latent variable models (Q626207)

A model is considered in which the latent (unobservable) variable is \[ Y_i=(Y_{i1},\dots,Y_{ip})'=X_i\beta+\Lambda Z_i+\varepsilon_i, \] where \(X_i\) is a vector of covariates, \(Z_i\) is a vector of (Gaussian) random effects, \(\beta\), \(\Lambda\) are matrices of unknown coefficients, and \(\varepsilon_i\) are Gaussian i.i.d errors. The observed response is \(W_i=(W_{i1},\dots,W_{ip})'\), \(W_{ij}=I\{Y_{ij}>0,\;Y_{ij}=\max_k(Y_{ik})\}\) if \(j=1,\dots,p-1\), \(W_{ip}=I\{Y_{ik}\leq 0 \;\forall k\}\). This is called the multidimensional probit factor analysis (MNPFA) model. Maximum likelihood estimation of \(\beta\) and \(\Lambda\) is discussed under suitable identifiability conditions. A Monte Carlo expectation-maximization (MCEM) algorithm is proposed for calibration of the estimates. Estimation of the standard errors of these estimates is discussed. Results of simulations are presented.