Statistical Detection Theory II

# Statistical Detection Theory II - Detection and...

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Detection and Estimation (Spring , 2009) S tatistical Detection Theory II NCTU EE P.1 Statistical Detection Theory (II) ~ Composite Hypothesis Testing Now we deal with unknown parameters. The pdf under H 0 or under H 1 or under both hypotheses may not be completely specified. Example: DC Level in WGN with Unknown Amplitude (A > 0) The NP detector decides H 1 if

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Detection & Estimation (Spring, 2009) Statistical Detection Theory II NCTU EE P.2 However, P D will depend on A. Over all possible detectors that have a given P FA , the NP detectors that decides H 1 if yields the highest P D for any value of A, as long as A > 0. Remark: When this type of test exists, it is called a uniformly most powerful ( UMP ) test. Unfortunately, UMP tests seldom exist.
Detection & Estimation (Spring, 2009) Statistical Detection Theory II NCTU EE P.3 If - < A < , we would obtain different tests for A positive and A negative. if A > 0 if A < 0 Since the value of A is unknown, the NP approach does not result in a unique test. Remark: 9 One-sided test : Two-sided test : 9 For a UMP test to exist, the parameter test must be one-sided. 9 When a UMP test does not exist, we may use the performance of the NP detector as an upper bound. 9 A detector that assumes perfect knowledge of an unknown parameter to design the NP detector is referred to as a clairvoyant detector . Example: DC Level in WGN with Unknown Amplitude When A can take on positive and negative values, the clairvoyant detector decides H 1 if This detector is unrealizable. However, it provides an upper bound on performance. Under H 0 , For A > 0,

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Detection & Estimation (Spring, 2009) Statistical Detection Theory II NCTU EE P.4 For A < 0, For a constant P FA , we choose γ - = - γ + Under H 1 , For A > 0, For A < 0,
Detection & Estimation (Spring, 2009) Statistical Detection Theory II NCTU EE P.5 For this example, we may try to decide H 1 if For this detector, it can be shown that ~ Composite Hypothesis Testing Approaches Two major approaches to composite hypothesis testing: Bayesian Approach : consider the unknown parameters as realizations of random variables and assign a prior pdf. Generalized Likelihood Ratio Test (GLRT)

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## This note was uploaded on 07/21/2009 for the course EE IEE5703 taught by Professor Sheng-jyhwang during the Spring '09 term at National Chiao Tung University.

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Statistical Detection Theory II - Detection and...

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