Estimating parameter confidence intervals in possibly misspecified parameter redundant models

Modelers are often faced with the dilemma of deciding how to parameterize a probability model. Too few parameters may yield a misspecified model with uninterpretable parameter estimates due to model misspecification, while too many parameters may yield a correctly specified model which fits the observed data with uninterpretable parameter estimates due to parameter redundancy. During the model development process, it is therefore likely that situations will arise where it is desirable to evaluate possibly misspecified or parameter redundant models. In the context of maximum likelihood estimation, it has been shown (see Ran and Hu, 2017, and Cole, 2020, for relevant reviews) that the presence of parameter redundancy corresponds to situations where the Fisher Information Matrix (FIM) (i.e., the covariance matrix of the log-likelihood per data record) does not have full rank. Local identifiability in maximum likelihood estimation often corresponds to checking if the Hessian of the log-likelihood (LL) has full rank (e.g., White, 1982, Theorem 3.1). Classical asymptotic theory (e.g., White, 1982) often assumes that both the FIM and LL Hessian are full rank in order to obtain analytic formulas for estimating parameter confidence intervals. In this presentation it is shown that analytic formulas for estimating confidence intervals for some but not all parameters can sometimes be obtained in the presence of parameter redundancy (i.e., without the assumption that the FIM has full rank). Some preliminary simulation studies are reported to illustrate the practical applications of the theoretical results.