Estimating Linear and Quadratic forms via Indirect Observations
(22.03.2017)Аннотация
In this paper, we further develop the approach, originating in 14
(<a href="/abs/1311.6765">arXiv:1311.6765</a>),20 (<a href="/abs/1604.02576">arXiv:1604.02576</a>), to "computation-friendly" hypothesis
testing and statistical estimation via Convex Programming. Specifically, we
focus on estimating a linear or quadratic form of an unknown "signal," known to
belong to a given convex compact set, via noisy indirect observations of the
signal. Most of the existing theoretical results on the subject deal with
precisely stated statistical models and aim at designing statistical inferences
and quantifying their performance in a closed analytic form. In contrast to
this descriptive (and highly instructive) traditional framework, the approach
we promote here can be qualified as operational -- the estimation routines and
their risks are yielded by an efficient computation. All we know in advance is
that under favorable circumstances to be specified below, the risk of the
resulting estimate, whether high or low, is provably near-optimal under the
circumstances. As a compensation for the lack of "explanatory power," this
approach is applicable to a much wider family of observation schemes than those
where "closed form descriptive analysis" is possible.
The paper is a follow-up to our paper 20 (<a href="/abs/1604.02576">arXiv:1604.02576</a>) dealing with
hypothesis testing, in what follows, we apply the machinery developed in this
reference to estimating linear and quadratic forms.