  Given a sample of independent and identically distributed random variables, a
novel nonparametric maximum entropy method is presented to estimate the
underlying continuous univariate probability density function (pdf). Estimates
are found by maximizing a log-likelihood function based on single order
statistics after transforming through a sequence of trial cumulative
distribution functions that iteratively improve using a Monte Carlo random
search method. Improvement is quantified by assessing the random variables
against the statistical properties of sampled uniform random data. Quality is
determined using an empirically derived scoring function that is scaled to be
sample size invariant. The scoring function identifies atypical fluctuations,
for which threshold values are set to define objective criteria that prevent
under-fitting as trial iterations continue to improve the model pdf, and,
stopping the iteration cycle before over-fitting occurs. No prior knowledge
about the data is required. An ensemble of pdf models is used to reflect
uncertainties due to statistical fluctuations in random samples, and the
quality of the estimates is visualized using scaled residual quantile plots
that show deviations from size-invariant statistics. These considerations
result in a tractable method that holistically employs key principles of random
variables and their statistical properties combined with employing orthogonal
basis functions and data-driven adaptive algorithms. Benchmark tests show that
the pdf estimates readily converge to the true pdf as sample size increases.
Robust results are demonstrated on several test probability densities that
include cases with discontinuities, multi-resolution scales, heavy tails and
singularities in the pdf, suggesting a generally applicable approach for
statistical inference.
