  We consider the problem of finding the subset of order statistics that
contains the most information about a sample of random variables drawn
independently from some known parametric distribution. We leverage
information-theoretic quantities, such as entropy and mutual information, to
quantify the level of informativeness and rigorously characterize the amount of
information contained in any subset of the complete collection of order
statistics. As an example, we show how these informativeness metrics can be
evaluated for a sample of discrete Bernoulli and continuous Uniform random
variables. Finally, we unveil how our most informative order statistics
framework can be applied to image processing applications. Specifically, we
investigate how the proposed measures can be used to choose the coefficients of
the L-estimator filter to denoise an image corrupted by random noise. We show
that both for discrete (e.g., salt-pepper noise) and continuous (e.g., mixed
Gaussian noise) noise distributions, the proposed method is competitive with
off-the-shelf filters, such as the median and the total variation filters, as
well as with wavelet-based denoising methods.
