Nonresponse is frequently encountered in empirical studies. When the response mechanism is missing not at random (MNAR) statistical inference using the observed data is quite challenging. Handling MNAR data often requires two model assumptions: one for the outcome and the other for the response propensity. Correctly specifying these two model assumptions is challenging and difficult to verify from the responses obtained. In this article we propose a semiparametric maximum likelihood method for MNAR data in the sense that a parametric assumption is used for the response propensity part of the model and a nonparametric model is used for the outcome part. The resulting analysis is more robust than the fully parametric approach. Some asymptotic properties of our estimators are derived. Results from a simulation study are also presented.