After the first four years of data taking, the IceCube neutrino telescope has observed 54 high-energy starting events (HESE) with deposited energies between 20 TeV and 2 PeV. The background from atmospheric muons and neutrinos is expected to be of about 20 events, all below 100 TeV, thus pointing towards the astrophysical origin of about 8 events per year in that data set. However, their precise origin remains unknown. Here, we perform a detailed analysis of this event sample (considering

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... aneously the energy, hemisphere and topology of the events) by assuming two contributions for the signal events: an isotropic power-law flux and a flux from decaying heavy dark matter. We fit the mass and lifetime of the dark matter and the normalization and spectral index of an isotropic power-law flux, for various decay channels of dark matter. We find that a significant contribution from dark matter decay is always slightly favored, either to explain the excess below 100 TeV, as in the case of decays to quarks or, as in the case of neutrino channels, to explain the three multi-PeV events. Also, we consider the possibility to interpret all the data by dark matter decays only, considering various combinations of two decay channels. We show that the decaying dark matter scenario provides a better fit to HESE data than the isotropic power-law flux.