  Statistical inference with nonresponse is quite challenging, especially when
the response mechanism is nonignorable. In this case, the validity of
statistical inference depends on untestable correct specification of the
response model. To avoid the misspecification, we propose semiparametric
Bayesian estimation in which an outcome model is parametric, but the response
model is semiparametric in that we do not assume any parametric form for the
nonresponse variable. We adopt penalized spline methods to estimate the unknown
function. We also consider a fully nonparametric approach to modeling the
response mechanism by using radial basis function methods. Using Polya-gamma
data augmentation, we developed an efficient posterior computation algorithm
via Gibbs sampling in which most full conditional distributions can be obtained
in familiar forms. The performance of the proposed method is demonstrated in
simulation studies and an application to longitudinal data.
