Limited and Full Information Estimation and Goodness-of-Fit Testing in 2n Contingency Tables

High-dimensional contingency tables tend to be sparse and standard goodness-of-fit statistics such as X2 cannot be used without pooling categories. As an improvement on arbitrary pooling, for goodness-of-fit of large 2n contingency tables, we propose a class of quadratic form statistics based on the residuals of margins or multivariate moments up to order r. Further, the marginal residuals are useful for diagnosing lack of fit of parametric models. These classes of test statistics are asymptotically chisquare and have better small sample properties than X2. We also show that these classes of test statistics have better power than X2 for some useful multivariate binary models. Related to this class of test statistics is a class of limited information estimators based on low-dimensional margins. We show that these estimators have high efficiency for one commonly used latent trait model for binary data