In this paper, we propose a novel spectral subtraction approach for speech enhancement via maximum likelihood estimate (MLE). This scheme attempts to simulate the probability distribution of useful speech signals and hence maximally reduce the noise. To evaluate the quality of speech enhancement, we extract cepstral features from the enhanced signals, and then apply them to a dynamic time warping framework for similarity check between the clean and filtered signals. The performance of the proposed enhancement method is compared to that of other classical techniques. The entire framework does not assume any model for the background noise and does not require any noise training data.