With a growing data privacy concern, federated learning has emerged as a promising framework to train machine learning models without sharing locally distributed data. In federated learning, local model training by multiple clients and model integration by a server are repeated only through model parameter sharing. Most existing federated learning methods assume training deep learning models, which are often computationally demanding. To deal with this issue, we propose federated learning methods with reservoir state analysis to seek computational efficiency and data privacy protection simultaneously. Specifically, our method relies on Mahalanobis Distance of Reservoir States (MD-RS) method targeting time series anomaly detection, which learns a distribution of reservoir states for normal inputs and detects anomalies based on a deviation from the learned distribution. Iterative updating of statistical parameters in the MD-RS enables incremental federated learning (IncFed MD-RS). We evaluate the performance of IncFed MD-RS using benchmark datasets for time series anomaly detection. The results show that IncFed MD-RS outperforms other federated learning methods with deep learning and reservoir computing models particularly when clients' data are relatively short and heterogeneous. We demonstrate that IncFed MD-RS is robust against reduced sample data compared to other methods. We also show that the computational cost of IncFed MD-RS can be reduced by subsampling from the reservoir states without performance degradation. The proposed method is beneficial especially in anomaly detection applications where computational efficiency, algorithm simplicity, and low communication cost are required.