Theory of Weak Identification in Semiparametric Models

We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, that is, when it is locally homogeneous of degree zero. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that there exists an underlying regular parameter that fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down a desirable one. If estimation of minimal sufficient underlying parameters is inefficient, it introduces noise in the corresponding estimation of weakly regular parameters, whence we can improve the estimators by local asymptotic Rao–Blackwellization. We call an estimator weakly efficient if it does not admit such improvement. New weakly efficient estimators are presented in linear IV and nonlinear regression models. Simulation of a linear IV model demonstrates how 2SLS and optimal IV estimators are improved.

Best integer equivariant estimation for elliptically contoured distributions

This contribution extends the theory of integer equivariant estimation (Teunissen in J Geodesy 77:402–410, 2003) by developing the principle of best integer equivariant (BIE) estimation for the class of elliptically contoured distributions. The presented theory provides new minimum mean squared error solutions to the problem of GNSS carrier-phase ambiguity resolution for a wide range of distributions. The associated BIE estimators are universally optimal in the sense that they have an accuracy which is never poorer than that of any integer estimator and any linear unbiased estimator. Next to the BIE estimator for the multivariate normal distribution, special attention is given to the BIE estimators for the contaminated normal and the multivariate t-distribution, both of which have heavier tails than the normal. Their computational formulae are presented and discussed in relation to that of the normal distribution.