  The device-independent approach to physics is one where conclusions are drawn
directly from the observed correlations between measurement outcomes. In
quantum information, this approach allows one to make strong statements about
the properties of the underlying systems or devices solely via the observation
of Bell-inequality-violating correlations. However, since one can only perform
a {\em finite number} of experimental trials, statistical fluctuations
necessarily accompany any estimation of these correlations. Consequently, an
important gap remains between the many theoretical tools developed for the
asymptotic scenario and the experimentally obtained raw data. In particular, a
physical and concurrently practical way to estimate the underlying quantum
distribution has so far remained elusive. Here, we show that the natural
analogs of the maximum-likelihood estimation technique and the
least-square-error estimation technique in the device-independent context
result in point estimates of the true distribution that are physical, unique,
computationally tractable and consistent. They thus serve as sound algorithmic
tools allowing one to bridge the aforementioned gap. As an application, we
demonstrate how such estimates of the underlying quantum distribution can be
used to provide, in certain cases, trustworthy estimates of the amount of
entanglement present in the measured system. In stark contrast to existing
approaches to device-independent parameter estimations, our estimation does not
require the prior knowledge of {\em any} Bell inequality tailored for the
specific property and the specific distribution of interest.
