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This paper is concerned with the moving horizon estimation (MHE) problem for linear discrete-time systems with limited communication, including quantized measurements and packet dropouts. The measured output is quantized by a logarithmic quantizer and the packet dropout phenomena is modeled by a binary switching random sequence. The main purpose of this paper is to design an estimator such that, for all possible quantized errors and packet dropouts, the state estimation error sequence is convergent. By choosing a stochastic cost function, the optimal estimator is obtained by solving a regularized least-squares problem with uncertain parameters. The proposed method can be used to deal with the estimation and prediction problems for systems with quantized errors and packet dropouts in a unified framework. The stability properties of the optimal estimator are also studied. The obtained stability condition implicitly establishes a relation between the upper bound of the estimation error and two parameters, namely, the quantization density and the packet dropout probability. Moreover, the maximum quantization density and the maximum packet dropout probability are given to ensure the convergence of the upper bound of the estimation error sequence. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]