TY - JOUR
T1 - A minimax Chebyshev estimator for bounded error estimation
AU - Eldar, Yonina C.
AU - Beck, Amir
AU - Teboulle, Marc
N1 - Funding Information:
Manuscript received April 11, 2007; revised June 24, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Erchin Serpedin. The work of Y. C. Eldar was supported by the Israel Science Foundation under Grant 536-04. The work of A. Beck and M. Teboulle was partially supported by the Israel Science Foundation Grant 489-06.
PY - 2008/4
Y1 - 2008/4
N2 - We develop a nonlinear minimax estimator for the classical linear regression model assuming that the true parameter vector lies in an intersection of ellipsoids. We seek an estimate that minimizes the worst-case estimation error over the given parameter set. Since this problem is intractable, we approximate it using semidefinite relaxation, and refer to the resulting estimate as the relaxed Chebyshev center (RCC). We show that the RCC is unique and feasible, meaning it is consistent with the prior information. We then prove that the constrained least-squares (CLS) estimate for this problem can also be obtained as a relaxation of the Chebyshev center, that is looser than the RCC. Finally, we demonstrate through simulations that the RCC can significantly improve the estimation error over the CLS method.
AB - We develop a nonlinear minimax estimator for the classical linear regression model assuming that the true parameter vector lies in an intersection of ellipsoids. We seek an estimate that minimizes the worst-case estimation error over the given parameter set. Since this problem is intractable, we approximate it using semidefinite relaxation, and refer to the resulting estimate as the relaxed Chebyshev center (RCC). We show that the RCC is unique and feasible, meaning it is consistent with the prior information. We then prove that the constrained least-squares (CLS) estimate for this problem can also be obtained as a relaxation of the Chebyshev center, that is looser than the RCC. Finally, we demonstrate through simulations that the RCC can significantly improve the estimation error over the CLS method.
KW - Bounded error estimation
KW - Chebyshev center
KW - Constrained least-squares
KW - Semidefinite programming
KW - Semidefinite relaxation
UR - http://www.scopus.com/inward/record.url?scp=41849151487&partnerID=8YFLogxK
U2 - 10.1109/TSP.2007.908945
DO - 10.1109/TSP.2007.908945
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AN - SCOPUS:41849151487
SN - 1053-587X
VL - 56
SP - 1388
EP - 1397
JO - IEEE Transactions on Signal Processing
JF - IEEE Transactions on Signal Processing
IS - 4
ER -