The statistical bias of numerically integrated statistical procedures (Q1070711)

Maximum likelihood estimation of multivariate normal models and Bayesian posterior density functions can require calculation using numerical integration. Although the statistical theory is unchanged for nonanalytical integrals, the exorbitant cost of estimating such models frequently induces researchers to use a small number of integration points. If the number of points is insufficient to calculate essentially exact values of the function being maximized, the resulting estimators are asymptotically biased and inconsistent both for the values of the estimated parameters and for their variances. The forms of the biases are derived and described in this paper.