Biased Cramer-Rao lower bound calculations for inequality-constrained estimators (Preprint)

Transcription

1 AFRL-DE-PS- JA AFRL-DE-PS- JA Biased Cramer-Rao lower bound calculations for inequality-constrained estimators (Preprint) Charles Matson Alim Haji 1 September 2006 Journal Article APPROVED FOR PUBLIC RELEASE; DISTRIBUTION IS UNLIMITED. AIR FORCE RESEARCH LABORATORY Directed Energy Directorate 3550 Aberdeen Ave SE AIR FORCE MATERIEL COMMAND KIRTLAND AIR FORCE BASE, NM

2 REPORT DOCUMENTATION PAGE Form Approved OMB No Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports ( ), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE 3. DATES COVERED (From - To) Oct 04-Aug 06 1 September 06 Journal Article 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER F D-0083 TO 06 Biased Cramer-Rao lower bound calculations for inequalityconstrained estimators (Preprint) 6. AUTHOR(S) Charles L. Matson, Alim Haji 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER F 5d. PROJECT NUMBER D0D0 5e. TASK NUMBER B3 5f. WORK UNIT NUMBER AD 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER AFRL/RDSA 3550 Aberdeen Avenue SE Kirtland AFB, NM SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) AFRL/RDSA Air Force Research Laboratory 3550 Aberdeen Ave. SE 11. SPONSOR/MONITOR S REPORT Kirtland AFB, NM NUMBER(S) AFRL-DE-PS-JA DISTRIBUTION / AVAILABILITY STATEMENT Approved for Public Release; Distribution is Unlimited. 13. SUPPLEMENTARY NOTES Submitted for publication in JOSA A, Vol. 23, No. 11, November GOVERNMENT PURPOSE RIGHTS 14. ABSTRACT Unbiased Cramer-Rao lower bound (CRB) theory can be used to calculate lower bounds to the variances of unbiased estimates of a set of parameters given only the probability density function of a random vector conditioned on the true parameter values. However, when the estimated parameter values are required to satisfy inequality constraints such as positivity, the resulting estimator is typically biased. To calculate CRBs for biased estimates of the parameter values, an expression for the bias gradient matrix must also be known. Unfortunately, this expression often does not exist. Because expressions for biased CRBs are preferable to sample variances calculations, alternate methods for deriving biased CRB expressions associated with inequality constraints are needed. Here we present an alternate approach that is biased upon creating the probability density function associated with a given biased estimate of these parameters using the available knowledge of estimator properties. We apply this approach to the calculations of biased CRBs for estimators that use a positivity constraint for a specific measurement model and discuss the benefits and limitations of this approach. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT Unclassified b. ABSTRACT Unclassified 18. NUMBER OF PAGES c. THIS PAGE Unclassified SAR 12 19a. NAME OF RESPONSIBLE PERSON Charles Matson 19b. TELEPHONE NUMBER (include area code) Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std

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