Practical framework for nonignorable response missingness with a nonparametric response component in partially linear mechanism models

We propose a practical likelihood-based semiparametric estimation method for generalised linear models with continuous responses subject to nonignorable missingness. The missingness mechanism is specified as a partially linear model: the probability of missingness is modelled nonparametrically as a function of a transformed response via regression splines, whereas the effects of covariates are specified parametrically. Identification is achieved by specifying the missingness mechanism through one of three monotonic link functions: logistic, probit, or complementary log-log, assuming the presence of a continuous covariate. Under suitable regularity conditions, the proposed estimators achieve asymptotic normality for the parametric components and attain the desired convergence rate for the nonparametric function. Simulation studies confirm robust performance across diverse scenarios, and the method is illustrated using data from the 2013 Chinese Household Income Project.