Estimation and inference with a (nearly) singular Jacobian

This paper develops extremum estimation and inference results for nonlinear models with very general forms of potential identification failure when the source of this identification failure is known. We examine models that may have a general deficient rank Jacobian in certain parts of the parameter space. When identification fails in one of these models, it becomes underidentified and the identification status of individual parameters is not generally straightforward to characterize. We provide a systematic reparameterization procedure that leads to a reparametrized model with straightforward identification status. Using this reparameterization, we determine the asymptotic behavior of standard extremum estimators and Wald statistics under a comprehensive class of parameter sequences characterizing the strength of identification of the model parameters, ranging from nonidentification to strong identification. Using the asymptotic results, we propose hypothesis testing methods that make use of a standard Wald statistic and data‐dependent critical values, leading to tests with correct asymptotic size regardless of identification strength and good power properties. Importantly, this allows one to directly conduct uniform inference on low‐dimensional functions of the model parameters, including one‐dimensional subvectors. The paper illustrates these results in three examples: a sample selection model, a triangular threshold crossing model, and a collective model for household expenditures.

Special configurations of robot-arms via screw theory

SUMMARYThe Jacobian of serial robot-arms is examined, and the matrix of cofactors of a singular Jacobian is presented as a means of explaining the physical nature of special configurations. Because the columns of both these matrices are screw coordinates, screw theory is central to proper understanding. ‘Realistic’ robot-arms are seen to behave in ways that can be explained not by particularizing from a general formulation but rather by carefully interpreting the relevant special screw systems from the outset. Higher singularities (with more than one freedom-loss) are then touched upon.