A Novel Nonparametric Maximum Likelihood Estimator for Probability Density Functions

Parametric maximum likelihood (ML) estimators of probability density functions (pdfs) are widely used today because they are efficient to compute and have several nice properties such as consistency, fast convergence rates, and asymptotic normality. However, data is often complex making parametrization of the pdf difficult, and nonparametric estimation is required. Popular nonparametric methods, such as kernel density estimation (KDE), produce consistent estimators but are not ML and have slower convergence rates than parametric ML estimators. Further, these nonparametric methods do not share the other desirable properties of parametric ML estimators. This paper introduces a nonparametric ML estimator that assumes that the square-root of the underlying pdf is band-limited (BL) and hence "smooth". The BLML estimator is computed and shown to be consistent. Although convergence rates are not theoretically derived, the BLML estimator exhibits faster convergence rates than state-of-the-art nonparametric methods in simulations. Further, algorithms to compute the BLML estimator with lesser computational complexity than that of KDE methods are presented. The efficacy of the BLML estimator is shown by applying it to (i) density tail estimation and (ii) density estimation of complex neuronal receptive fields where it outperforms state-of-the-art methods used in neuroscience.