In this work, we propose a framework for supervisory cooperative estimation of multi-agent linear time-invariant (LTI) systems. We introduce a group of sub-observers, each estimating certain states that are conditioned on given input, output, and state information. The cooperation among the sub-observers is supervised by a discrete-event system (DES) supervisor. The supervisor makes decisions on selecting and configuring a set of sub-observers to successfully estimate all states of the system. Moreover, when certain anomalies are present, the supervisor reconfigures the set of selected sub-observers so that the impact of anomalies on the estimation performance is minimized. This framework is applicable to any multi-agent system including large-scale industrial processes. In this paper (Part I), our proposed framework for supervisory estimation is developed based on the notion of sub-observers and DES supervisory control. In the companion paper (Part II), a DES-based combinatorial optimization method for selection of an optimal set of sub-observers is presented, the feasibility of the overall integrated sub-observers is validated, and the application of our proposed method in a practical industrial process is demonstrated through numerical simulations.