  Nonresponse after probability sampling is a universal challenge in survey
sampling, often necessitating adjustments to mitigate sampling and selection
bias simultaneously. This study explored the removal of bias and effective
utilization of available information, not just in nonresponse but also in the
scenario of data integration, where summary statistics from other data sources
are accessible. We reformulate these settings within a two-step monotone
missing data framework, where the first step of missingness arises from
sampling and the second originates from nonresponse. Subsequently, we derive
the semiparametric efficiency bound for the target parameter. We also propose
adaptive estimators utilizing methods of moments and empirical likelihood
approaches to attain the lower bound. The proposed estimator exhibits both
efficiency and double robustness. However, attaining efficiency with an
adaptive estimator requires the correct specification of certain working
models. To reinforce robustness against the misspecification of working models,
we extend the property of double robustness to multiple robustness by proposing
a two-step empirical likelihood method that effectively leverages empirical
weights. A numerical study is undertaken to investigate the finite-sample
performance of the proposed methods. We further applied our methods to a
dataset from the National Health and Nutrition Examination Survey data by
efficiently incorporating summary statistics from the National Health Interview
Survey data.
