We report on a study of the photoluminescence of a series of indium-doped n-type CdSe samples with carrier concentrations above the critical concentration for the metal-insulator transition. The photoluminescence was measured for excitation energies both greater and smaller than the optical gap, and excitation spectroscopy was also used as a probe. As in earlier results for CdS the spectra of CdSe exhibit a sharp edge at high energies corresponding to the decay of electrons from the Fermi level in the conduction or donor-impurity band to states in the valence band, and a broad tail at lower energies due to transitions from the rest of the Fermi sea. While in CdS the high-energy edge moved monotonically to higher energy with increasing carrier concentration, N(D)-N(A), this monotonic behavior did not hold for two of the CdSe samples. We suggest that this unexpected behavior can be attributed to the effect of different levels of compensation among the samples used in the present studies. The full results are interpreted in terms of band filling, electronic interactions, and band-gap renormalization in the presence of compensation.