In this paper, we propose a general framework to estimate short-time spectral amplitudes (STSA) of speech signals in noise by joint speech detection and estimation to remove or reduce background noise, without increasing signal distortion. The approach is motivated by the fact that speech signals have sparse time-frequency representations and can reasonably be assumed not to be present in every time-frequency bin of the time-frequency domain. By combining parametric detection and estimation theories, the main idea is to take into consideration speech presence and absence in each time-frequency bin to improve the performance of Bayesian estimators. In this respect, for three Bayesian estimators, optimal Neyman-Pearson detectors are derived to decide on the absence or presence of speech in each given time-frequency bin. Decisions returned by such detectors are then used to improve the initial estimates. The resulting estimations have been assessed in two scenarios, namely, with and without reference noise power spectrum. The objective tests confirm the relevance of these approaches, both in terms of speech quality and intelligibility.