Standard Errors for Maximum Likelihood Estimates in Possibly Misspecified Parameter Redundant Models

In many applications, one is interested in estimating standard errors for maximum likelihood estimates but it is typically assumed that such standard errors are not valid if the covariance matrix of the maximum likelihood estimates is singular. Such a situation can arise, for example, if there are more parameters than data points in the model. In this talk, we show how it is sometimes possible to estimate standard errors for maximum likelihood estimates in the simultaneous presence of parameter redundancy and possible model misspecification. The essential idea is that even if the covariance matrix of the maximum likelihood estimates is singular, it may still be possible to show that certain linear combinations of the maximum likelihood estimates have a legitimate asymptotic Gaussian distribution. The theory is applicable to a broad class of smooth nonlinear finite-dimensional parametric probability models including exponential family models and latent variable models. After sketching the main theorem, a practical algorithm is developed and applied to the analysis of standard error estimation for a Deterministic Input Noisy And (DINA) Cognitive Diagnostic Model (CDM) fit to an extract of the Tatsuoka (1983) Fraction-Subtraction data set. Simulation studies are also reported to help evaluate the validity of the asymptotic approximation.