We propose two novel voice activity detection (VAD) algorithms by employing i) the Rényi entropy estimate, and ii) a weighted combination of Rényi and differential entropies, at each frequency bin of power spectral estimates, by considering overlapping frames in the received signal samples. The Bartlett-Welch method is employed to estimate the power spectrum from a suitably chosen number of past frames in the frequency domain. Exploiting the known fact that a long-term averaging enhances the performance of VAD, we first propose a novel frequency domain long-term quadratic Rényi entropy (FLQRE) feature, which is extracted by summing the estimated Rényi entropy values across the frequency bins over each signal frame. Later, we propose a weighted combination of the frequency domain long-term differential entropy (FLDE) and FLQRE feature for VAD, termed as a weighted quadratic and differential entropy (WQDE) feature. We evaluate the performance of the proposed FLQRE and WQDE VAD schemes, by considering twelve types of noises from NOISEX-92 at five different SNR values, artificially added to speech samples from SWITCHBOARD and TIMIT corpora. We present an extensive performance comparison study to establish the utility of our proposed VAD algorithms over state-of-the-art short- and long-term VAD techniques such as ITU-T G.729B, likelihood ratio test, long-term signal variability, and long-term spectral flatness measure-based algorithms. We show that the proposed algorithms yield the best average accuracy and noise-hit rate under the SWITCHBOARD corpus, and yield a comparable performance under the TIMIT corpus.