Asymptotic cumulants of the Akaike and Takeuchi information criteria are given under possible model misspecification up to the fourth order with the higher-order asymptotic variances, where two versions of the latter information criterion are defined using observed and estimated expected information matrices. The asymptotic cumulants are provided before and after studentization using the parameter estimators by the weighted-score method, which include the maximum likelihood and Bayes modal estimators as special cases. Higher-order bias corrections of the criteria are derived using log-likelihood derivatives, which yields simple results for cases under canonical parametrization in the exponential family. It is shown that in these cases the Jeffreys prior gives the vanishing higher-order bias of the Akaike information criterion. The results are illustrated by three examples. Simulations for model selection in regression and interval estimation are also given.

In this paper, we consider the determination of the number of factors in nonnegative matrix factorization (NMF) for a zero-inflated data matrix. This zero-inflated case leads to poor approximation to the nonnegative data matrix. To address this problem, we use the zero-inflated compound Poisson-gamma distribution as the error distribution in NMF. In addition, we consider automatic relevance determination (ARD) for model order selection. Our simulation study shows that our method is better than the basic ARD method for zero-inflated data. We apply our proposed method to real-world purchasing data to determine the number of buying patterns.