This paper presents an algorithm for camera localization using trajectory estimation (CLUTE) in a distributed network of nonoverlapping cameras. The algorithm recovers the extrinsic calibration parameters, namely, the relative position and orientation of the camera network on a common ground plane coordinate system. We first model the observed trajectories in each camera's field of view using Kalman filtering, then we use this information to estimate the missing trajectory information in the unobserved areas by fusing the results of a forward and backward linear regression estimation from adjacent cameras. These estimated trajectories are then filtered and used to recover the relative position and orientation of the cameras by analyzing the estimated and observed exit and entry points of an object in each camera's field of view. The final configuration of the network is established by considering one camera as a reference and by adjusting the remaining cameras with respect to this reference. We demonstrate the algorithm on both simulated and real data and compare the results with state-of-the-art approaches. The experimental results show that the proposed algorithm is more robust to noisy and missing data and in case of camera failure.