ECE 561: Detection and Estimation Theory
Spring 2009
Issued: April 8, 2009
April 8, 2009
•1)
Read the treatment of asymptotic robustness and robust detection on
pages 250-263 of Levy, and provide a concise, one-or-two page summary
of the most important results in the text.
•2)
Under which conditions on the pdf’s and loss function does there exist
a natural ordering among the performances of MMSE, MAE, and MAP
estimators (in class, we showed what the conditions are for all three to be
identical. This questions asks you to establish when can you claim one to
be strictly larger than the other two, for all possible pairs and one triple).
•3)
Consider the modified LLS problem, in which you seek an estimator of
Xof the form ˆ
X=AY+b, but with Arequired to be upper or lower
triangular. As an example, you could also consider another form of a LLS
estimator with structural constraints on A. How would you solve this
problem? When does the LLS estimator allow for Ato be triangular?
•4)
Prove the following statements about the Schur complement D/M of a
matrix Dwith respect to the block matrix M, where Mconsists of blocks
A, B, C, D, respectively:
a) det(M) = det(D)×det(D/M).
b) The inertia of a Hermitian matrix Mis an ordered triple, IM=
(m>, m<, m0), where m>,m<, and m0denote the number of positive,
negative, and zero-valued eigenvalues of M. Show that
IM=ID+ID/M .
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