31 research outputs found
Non-Bayesian Post-Model-Selection Estimation as Estimation Under Model Misspecification
In many parameter estimation problems, the exact model is unknown and is
assumed to belong to a set of candidate models. In such cases, a predetermined
data-based selection rule selects a parametric model from a set of candidates
before the parameter estimation. The existing framework for estimation under
model misspecification does not account for the selection process that led to
the misspecified model. Moreover, in post-model-selection estimation, there are
multiple candidate models chosen based on the observations, making the
interpretation of the assumed model in the misspecified setting non-trivial. In
this work, we present three interpretations to address the problem of
non-Bayesian post-model-selection estimation as an estimation under model
misspecification problem: the naive interpretation, the normalized
interpretation, and the selective inference interpretation, and discuss their
properties. For each of these interpretations, we developed the corresponding
misspecified maximum likelihood estimator and the misspecified
Cramr-Rao-type lower bound. The relations between the
estimators and the performance bounds, as well as their properties, are
discussed. Finally, we demonstrate the performance of the proposed estimators
and bounds via simulations of estimation after channel selection. We show that
the proposed performance bounds are more informative than the oracle
Cramr-Rao Bound (CRB), where the third interpretation
(selective inference) results in the lowest mean-squared-error (MSE) among the
estimators.Comment: This work has been submitted to the IEEE for possible publication.
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Widely-Linear MMSE Estimation of Complex-Valued Graph Signals
In this paper, we consider the problem of recovering random graph signals
with complex values. For general Bayesian estimation of complex-valued vectors,
it is known that the widely-linear minimum mean-squared-error (WLMMSE)
estimator can achieve a lower mean-squared-error (MSE) than that of the linear
minimum MSE (LMMSE) estimator. Inspired by the WLMMSE estimator, in this paper
we develop the graph signal processing (GSP)-WLMMSE estimator, which minimizes
the MSE among estimators that are represented as a two-channel output of a
graph filter, i.e. widely-linear GSP estimators. We discuss the properties of
the proposed GSP-WLMMSE estimator. In particular, we show that the MSE of the
GSP-WLMMSE estimator is always equal to or lower than the MSE of the GSP-LMMSE
estimator. The GSP-WLMMSE estimator is based on diagonal covariance matrices in
the graph frequency domain, and thus has reduced complexity compared with the
WLMMSE estimator. This property is especially important when using the
sample-mean versions of these estimators that are based on a training dataset.
We then state conditions under which the low-complexity GSP-WLMMSE estimator
coincides with the WLMMSE estimator. In the simulations, we investigate two
synthetic estimation problems (with linear and nonlinear models) and the
problem of state estimation in power systems. For these problems, it is shown
that the GSP-WLMMSE estimator outperforms the GSP-LMMSE estimator and achieves
similar performance to that of the WLMMSE estimator.Comment: This work has been submitted to the IEEE for possible publication.
Copyright may be transferred without notice, after which this version may no
longer be accessibl