  When the response mechanism is believed to be not missing at random (NMAR), a
valid analysis requires stronger assumptions on the response mechanism than
standard statistical methods would otherwise require. Semiparametric estimators
have been developed under the model assumptions on the response mechanism. In
this paper, a new statistical test is proposed to guarantee model
identifiability without using any instrumental variable. Furthermore, we
develop optimal semiparametric estimation for parameters such as the population
mean. Specifically, we propose two semiparametric optimal estimators that do
not require any model assumptions other than the response mechanism. Asymptotic
properties of the proposed estimators are discussed. An extensive simulation
study is presented to compare with some existing methods. We present an
application of our method using Korean Labor and Income Panel Survey data.
