Although angle random walk (ARW) of fiber optic gyroscope (FOG) has been well modeled and identified before being integrated into the high-accuracy attitude control system of satellite, aging and unexpected failures can affect the performance of FOG after launch, resulting in the variation of ARW coefficient. Therefore, the ARW coefficient can be regarded as an indicator of "state of health" for FOG diagnosis in some sense. The Allan variance method can be used to estimate ARW coefficient of FOG, however, it requires a large amount of data to be stored. Moreover, the procedure of drawing slope lines for estimation is painful. To overcome the barriers, a weighted state-space model that directly models the ARW to obtain a nonlinear state-space model was established for FOG. Then, a neural extended-Kalman filter algorithm was implemented to estimate and track the variation of ARW in real time. The results of experiment show that the proposed approach is valid to detect the state of FOG. Moreover, the proposed technique effectively avoids the storage of data.