New strategies for detecting atypical observations based on the information matrix equality

Diagnostic analyses in regression modeling are usually based on residuals or local influence measures and are used for detecting atypical observations. We develop a new approach for identifying such observations when the parameters of the model are estimated by maximum likelihood. The proposed approach is based on the information matrix equality, which holds when the model is correctly specified. We introduce a new definition of an atypical observation: one that disproportionately affects the degree of adequate specification of the model as measured using the sample counterparts of the matrices that comprise the information matrix equality. We consider various measures of distance between two symmetric matrices and apply them such that a zero distance indicates correct model specification. These measures quantify the degree of model adequacy and help identify atypical cases that significantly impact the model's adequacy. We also introduce a modified generalized Cook distance and a new criterion that uses the two generalized Cook's distances (modified and unmodified). Empirical applications involving Gaussian and beta regression models are presented and discussed.