  Covariate balancing methods have been widely applied to single or monotone
missing patterns and have certain advantages over likelihood-based methods and
inverse probability weighting approaches based on standard logistic regression.
In this paper, we consider non-monotone missing data under the complete-case
missing variable condition (CCMV), which is a case of missing not at random
(MNAR). Using relationships between each missing pattern and the complete-case
subsample, a weighted estimator can be used for estimation, where the weight is
a sum of ratios of conditional probability of observing a particular missing
pattern versus that of observing the complete-case pattern, given the variables
observed in the corresponding missing pattern. Plug-in estimators of the
propensity ratios, however, can be unbounded and lead to unstable estimation.
Using further relations between propensity ratios and balancing of moments
across missing patterns, we employ tailored loss functions each encouraging
empirical balance across patterns to estimate propensity ratios flexibly using
functional basis expansion. We propose two penalizations to separately control
propensity ratio model complexity and covariate imbalance. We study the
asymptotic properties of the proposed estimators and show that they are
consistent under mild smoothness assumptions. Asymptotic normality and
efficiency are also developed. Numerical simulation results show that the
proposed method achieves smaller mean squared errors than other methods.
