A new algorithm is developed to train feed-forward neural networks for non-linear input-to-output mappings with small incomplete data in arbitrary distributions. The developed Training-EStimation-Training (TEST) algorithm consists of 3 steps, i.e., (1) training with the complete portion of the training data set, (2) estimation of the missing attributes with the trained neural networks, and (3) re-training the neural networks with the whole data set. Error back propagation is still applicable to estimate the missing attributes. Unlike other training methods with missing data, it does not assume data distribution models which may not be appropriate for small training data. The developed TEST algorithm is first tested for the Iris benchmark data. By randomly removing some attributes from the complete data set and estimating the values latter, accuracy of the TEST algorithm is demonstrated. Then it is applied to the Diabetes benchmark data, of which about 50% contains missing attributes. Compared with other existing algorithms, the proposed TEST algorithm results in much better recognition accuracy for test data.