Abstract We consider nonparametric complex spectral estimation of data sequences with missing samples occurring in arbitrary patterns. The existing spectral estimation algorithms designed for uniformly sampled complete-data sequences perform poorly when applied to data sequences with missing samples if the missing samples are simply set to zero. Several nonparametric algorithms have recently been developed to deal with the missing-data problem. They include, for example, GAPES for gapped data and PG-APES, PG-CAPON for periodically gapped data. However, they are not really suitable for the general missing-data problem where the missing data samples occur in arbitrary patterns. In this paper, we deal with a general missing-data spectral estimation problem for which we develop two nonparametric missing-data amplitude and phase estimation (MAPES) algorithms, both of which make use of the expectation maximization (EM) algorithm. Numerical results are provided to demonstrate the effectiveness of the proposed algorithms.