Hypothesis Testing. Presenting: Lihu Berman. Agenda. Basic concepts Neyman-Pearson lemma UMP Invariance CFAR. X is a random vector with distribution. is a parameter, belonging to the parameter space. disjoint covering of the parameter space. denotes the hypothesis that.
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is said to be a simple hypothesis
simple vs. composite hypotheses
A two-sided test: the alternative lies on both sides of
A one-sided test (for scalar ):Basic concepts (cont.)
Otherwise, it is said to be a composite hypothesis
RADAR – Is there a target or not ?
Physical model – is the coin we are tossing fair or not ?
Natural logarithm is monotone, enabling the use of Log-Likelihood
Assume equal energies: and define
making the most powerful test of size for testing
the composite alternative vs. the simple hypothesis
Consider now the power function
At . For any because is
more powerful than the test
A similar argument holds for anyUMP Tests (cont.)
Fisher-Neyman factorization theorem:
The statistic T(x) is sufficient for if and only ifA note on sufficiency
No other statistic which can be calculated from the same sample
provides any additional information as to the value of the parameter
One can write the likelihood-ratio in terms of the sufficient statistic
So what?! Let us continue with the log-likelihood as before…
Intuitively: search for a statistic that is invariant to the nuisance parameterInvariance (cont.)
Project the measurement on the subspace orthogonal to the disturbance!
The angle between the measurement and the signal-subspace
(or the subspace orthogonal to it: )
In fact, Z is a maximal invariant statistic to
a broader group G’, that includes also
rotation in the subspace.
G’ is specifically appropriate for channels
that introduce rotation in as well as gainInvariance (another example)
1. Find a meaningful group of transformations, for which the hypothesis testing problem is invariant.
2. Find a maximal invariant statistic M, and construct a likelihood ratio test.
3. If M has a monotone likelihood ratio, then the test is UMPI for testing one sided hypotheses of the form
Project the measurement on the signal space. A UMP test !
The False-Alarm Rate is Constant. Thus: CFAR
Redraw the problem as:CFAR (cont.)
Utilize Invariance !!
is unknown !!!
Thus, we can set a threshold in the test:
in order to obtain
Furthermore, as the likelihood ratio for non-central t is monotone, this
test is UMPI for testing in the distribution
when is unknown !CFAR (cont.)
The actual probability of detection depends on the actual value of the SNR