This paper is concerned with the identification problems of the linear parameter varying (LPV) system with missing output in the presence of the time-delay. A multiple-model approach is adopted. Local models varying from one operating point to another are first described by finite impulse response (FIR) models. To handle missing output and time-delay, the expectation-maximization (EM) algorithm is utilized to estimate the unknown parameters and the time-delay simultaneously. Output Error (OE) models are widely used in controller design. Therefore, the auxiliary model principle is employed to recover the OE models based on the initially identified FIR models. The EM algorithm is then used again to refine the unknown parameters of the OE models with the complete data set to obtain the final global model. Simulation examples are presented to demonstrate the performance of the proposed method.