Close to glottal closure instants (GCIs), the speech signal is expected to change its amplitude rapidly and, at GCIs, it is expected to have strong negative peaks. A novel algorithm that exploits these two properties for the estimation of GCIs is presented. Here, a symmetrised speech segment is assumed to be a Fourier transform (FT) of an even function. In such a case, at the locations of the GCIs, the strong negative peaks in the symmetrised speech segment correspond to zeros that lie considerably outside the unit circle in the z-plane. The group delay spectrum of the time-domain signal derived by taking inverse FT of this assumed FT is expected to take a value close to −2π at the angular locations of these zeros. Mapping frequency scale to time scale, the frequency bins for which group delay reaches −2π correspond to the locations of GCIs. Theoretical justification for the proposed approach is also presented by defining a novel function called the conditional group delay function. Systematic evaluation is carried out on the CMU Arctic database and the performance of the proposed technique is better than that of the algorithms namely DYPSA, ZFF, YAGA and is close to that of SEDREAMS.