Estimation with multivariate outcomes having nonignorable item nonresponse

To estimate unknown population parameters based on \({\varvec{y}}\), a vector of multivariate outcomes having nonignorable item nonresponse that directly depends on \({\varvec{y}}\), we propose an innovative inverse propensity weighting approach when the joint distribution of \({\varvec{y}}\) and associated covariate \({\varvec{x}}\) is nonparametric and the nonresponse probability conditional on \({\varvec{y}}\) and \({\varvec{x}}\) has a parametric form. To deal with the identifiability issue, we utilize a nonresponse instrument \({\varvec{z}}\), an auxiliary variable related to \({\varvec{y}}\) but not related to the nonresponse probability conditional on \({\varvec{y}}\) and \({\varvec{x}}\). We utilize a modified generalized method of moments to obtain estimators of the parameters in the nonresponse probability. Simulation results are presented and an application is illustrated in a real data set.