  We consider the statistical inference for noisy incomplete binary (or 1-bit)
matrix. Despite the importance of uncertainty quantification to matrix
completion, most of the categorical matrix completion literature focuses on
point estimation and prediction. This paper moves one step further toward the
statistical inference for binary matrix completion. Under a popular nonlinear
factor analysis model, we obtain a point estimator and derive its asymptotic
normality. Moreover, our analysis adopts a flexible missing-entry design that
does not require a random sampling scheme as required by most of the existing
asymptotic results for matrix completion. Under reasonable conditions, the
proposed estimator is statistically efficient and optimal in the sense that the
Cramer-Rao lower bound is achieved asymptotically for the model parameters. Two
applications are considered, including (1) linking two forms of an educational
test and (2) linking the roll call voting records from multiple years in the
United States Senate. The first application enables the comparison between
examinees who took different test forms, and the second application allows us
to compare the liberal-conservativeness of senators who did not serve in the
Senate at the same time.
