In this paper, we design a slow-rate batch form and a fast-rate iterative form of the finite impulse response (FIR) filter based on the law of great likelihood for the state space model with time delay and random observation loss obeying Bernoulli distribution. Firstly, the model in the case of time delay and data loss is formulated as a linear function of probability obeying Bernoulli distribution, and then the proposed FIR algorithm with great likelihood is obtained by the great likelihood process. Finally, the maximum likelihood FIR estimation, the improved Kalman filter and the unbiased FIR estimation under the same conditions are compared and analyzed in terms of estimation error, root mean square error and uncertainty impact. In the experimental part, the 3-DOF helicopter model simulation shows that the proposed maximum likelihood FIR estimation method is more effective and robust in dealing with time delay and data loss problems.