Efficiency Bound of Local Z-Estimators on Discrete Sample Spaces

Many statistical models over a discrete sample space often face the computational difficulty of the normalization constant. Because of that, the maximum likelihood estimator does not work. In order to circumvent the computation difficulty, alternative estimators such as pseudo-likelihood and composite likelihood that require only a local computation over the sample space have been proposed. In this paper, we present a theoretical analysis of such localized estimators. The asymptotic variance of localized estimators depends on the neighborhood system on the sample space. We investigate the relation between the neighborhood system and estimation accuracy of localized estimators. Moreover, we derive the efficiency bound. The theoretical results are applied to investigate the statistical properties of existing estimators and some extended ones.