Finding anomalies using moving average in a time series

Basic anomaly detection can be done looking for the divergence between a short-term moving average and a long-term. This is very crude, however.

If you are looking for anomalies, you can do the Moving Average, and then use a threshold to detect passings of this. This is crude, and if you care about delays it will also impact you in that sense.

I do not know what you mean by estimate the time series?

To detect anomalies, I would use other things, like wavelet transform,which tend to be pretty good for this.

The idea to find anomalies in time series is:

1) Create a model to predict each next sample, or state. (Say a LMS, or Kernel-LMS filter, maybe a Kalman Filter or Gaussian Processes Regressor)
2) Quantify how weird (quantify error, novelty or surprise) is this new sample to your model.
3) Set a threshold and decide if the new sample is an anomaly or not.
4) Adapt your model based on your decision and analyze the sample (return to 2)

My lab had interesting results on this subject. Check it out this paper: http://cnel.ufl.edu/files/1317347633.pdf