In this paper, we address the problem of simultaneously detecting and estimating the speech short-time spectral amplitude (STSA) through combinations of the Neyman-Pearson and Bayesian approaches for speech enhancement. The main idea is that a non-continuous cost function, which depends on the absence/presence of speech, is applied to optimal joint detection and estimation for improving performance of mean square error estimators. Furthermore, our proposed method based on the square error of the logarithmic STSA makes it possible for us to reduce much more the background noise without introducing severe signal distortion. Preliminary experimental results demonstrate the advantage of our method in terms of speech quality and intelligibility