The estimation of a density profile from experimental data points is a
challenging problem, usually tackled by plotting a histogram. Prior assumptions
on the nature of the density, from its smoothness to the specification of its
form, allow the design of accurate estimation procedures, such as Maximum
Likelihood. Our aim is to construct a procedure that makes the smallest
possible number of assumptions, but still providing an accurate estimate of the
density. We introduce the self-consistent estimate: the power spectrum of a
candidate density is given, and an estimation procedure is performed on the
assumption, to be released a posteriori, that the candidate is correct. The
self-consistent estimate is defined as a prior candidate density that precisely
reproduces itself. Our main result is to show that the self-consistent estimate
is unique, for any given dataset, and to derive its exact analytic expression.
Applications of the method 1) do not require any assumption about the form of
the density, and 2) do not depend on the subjective choice of any adjustable
parameter, such as a bin size, a kernel bandwidth or a cutoff frequency. We
study its application to Gaussian and Cauchy distributions: although the
self-consistent estimate is non-parametric, it reaches the theoretical limit of
Maximum Likelihood for the scaling of the square error with the dataset size.