  In this paper, a nonparametric maximum likelihood (ML) estimator for
band-limited (BL) probability density functions (pdfs) is proposed. The BLML
estimator is consistent and computationally efficient. To compute the BLML
estimator, three approximate algorithms are presented: a binary quadratic
programming (BQP) algorithm for medium scale problems, a Trivial algorithm for
large-scale problems that yields a consistent estimate if the underlying pdf is
strictly positive and BL, and a fast implementation of the Trivial algorithm
that exploits the band-limited assumption and the Nyquist sampling theorem
("BLMLQuick"). All three BLML estimators outperform kernel density estimation
(KDE) algorithms (adaptive and higher order KDEs) with respect to the mean
integrated squared error for data generated from both BL and infinite-band
pdfs. Further, the BLMLQuick estimate is remarkably faster than the KD
algorithms. Finally, the BLML method is applied to estimate the conditional
intensity function of a neuronal spike train (point process) recorded from a
rat's entorhinal cortex grid cell, for which it outperforms state-of-the-art
estimators used in neuroscience.
