1. Presenting: Lihu Berman
Hypothesis Testing
Presenting:
Lihu Berman

2. Agenda
Basic concepts
Neyman-Pearson lemma
UMP
Invariance
CFAR

3. Basic concepts 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
Binary test of hypotheses:
vs.
M-ary test:
vs.

4. Basic concepts (cont.) If then is said to be a simple hypothesis
Otherwise, it is said to be a composite hypothesis
Example:
vs.
simple vs. composite hypotheses
RADAR – Is there a target or not ?
Physical model – is the coin we are tossing fair or not ?
A two-sided test: the alternative lies on both sides of
A one-sided test (for scalar ):
vs.

5. Basic concepts (cont.)
Introduce the test function for binary test of hypotheses:
is a disjoint covering of the measurement space
If the measurement is in the Acceptance region – is accepted
If it is in the Rejection region – is rejected, and is accepted.

6. Basic concepts (cont.) Probability of False-Alarm (a.k.a. Size):
simple:
composite:
i.e. the worst case
Detection probability (a.k.a. Power):
simple:
composite:

7. Basic concepts (cont.) Receiving Operating Characteristics (ROC):
Chance line
The best test of size has the largest among all tests of that size

8. The Neyman-Pearson lemma
and let denote the density function of X, then:
Is the most powerful test of size for testing which of the two
simple hypotheses is in force, for some
Let:

9. The Neyman-Pearson (cont.)
Proof: Let denote any test satisfying:
Obviously:

10. The Neyman-Pearson (cont.)
Note 1: If then the most powerful test is:
Note 2: Introduce the likelihood function:
Then the most powerful test can be written as:

11. The Neyman-Pearson (cont.)
Note 3: Choosing the threshold k.
Denote by the probability density of the likelihood function
under , then:
Note 4: If is not continuous (i.e )
Then the previous equation might not work! In that case, use the test:
Toss a coin, and
choose if heads up

12. Binary comm. in AWGN Source Mapper
‘1’ = Enemy spotted. ‘0’ = All is clear.
Prior probabilities unknown !

13. Binary comm. in AWGN
Natural logarithm is monotone, enabling the use of Log-Likelihood

14. Binary comm. (cont.)

15. Binary comm. (cont.)
Assume equal energies: and define

16. Binary comm. (cont.)

17. Binary comm. (cont.)

18. Binary comm. (cont.)

19. UMP Tests The Neyman-Pearson lemma holds for simple hypotheses.
Uniformly Most Powerful tests generalize to composite hypotheses
A test is UMP of size , if for any other test , we have:

20. UMP Tests (cont.)
Consider scalar R.Vs whose PDFs are parameterized by scalar
If the likelihood-ratio is monotone non-decreasing in x for any
pair
Karlin-Rubin Theorem (for UMP one-sided tests):
Is the UMP test of size for testing
, then the test:

21. UMP Tests (cont.) Proof: begin with fixed values
By the Neyman-Pearson lemma, the most powerful test of size for
testing is:
As likelihood is monotone, we may replace it with the threshold test

22. UMP Tests (cont.)
The test is independent of , so the argument holds for every
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 any

23. UMP Tests (cont.) Thus, we conclude that is non-decreasing
Consequently, is also a test whose size satisfies
Finally, no test with size can have power , as it would
contradict Neyman-Pearson, in

24. A note on sufficiency
The statistic T(x) is sufficient for if and only if
No other statistic which can be calculated from the same sample
provides any additional information as to the value of the parameter
Fisher-Neyman factorization theorem:
The statistic T(x) is sufficient for if and only if
One can write the likelihood-ratio in terms of the sufficient statistic

25. UMP Tests (cont.) UMP one-sided tests exist for a host of problems !
Theorem: the one-parameter exponential family of distributions with
density:
has a monotone likelihood ratio in the sufficient statistic
provided that is non-decreasing
Proof:

26. UMP Tests (cont.)
Example:

27. UMP Tests (cont.) Therefore, the test
is the Uniformly Most Powerful test of size for testing

28. Invariance
Revisit the binary communication example, but with a slight change.
Source
Mapper
So what?! Let us continue with the log-likelihood as before…
Oops

29. Invariance (cont.)
Intuitively: search for a statistic that is invariant to the nuisance parameter
Project the measurement on the subspace orthogonal to the disturbance!
Optimal
signals ?

30. Invariance (formal discussion)
Let G denote a group of transformations.
X has probability distribution:

31. Invariance (cont.)
Revisit the previous example (AWGN channel with unknown bias)
The measurement is distributed as

32. Invariance (cont.)
organizes the measurements x into equivalent classes where:

33. Invariance (cont.)

34. Invariance (cont.)
Let us show that is indeed a maximal invariant statistic

35. Invariance (another example)
Consider the group of transformations:
The hypothesis test problem is invariant to G

36. Invariance (another example)
What statistic is invariant to the scale of S ?
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 gain

37. Invariance (UMPI & summary)
Invariance may be used to compress measurements into statistics of low dimensionality, that satisfy invariance conditions.
It is often possible to find a UMP test within the class of invariant tests.
Steps when applying invariance:
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
Note: Sufficiency principals may facilitate this process.

38. CFAR (introductory example)
Project the measurement on the signal space. A UMP test !
The False-Alarm Rate is Constant Thus: CFAR

39. CFAR (cont.)
m depends now on the unknown Test is useless. Certainly not CFAR
Redraw the problem as:
Utilize Invariance !!

40. CFAR (cont.)
As before:
Change slightly:
independent

41. CFAR (cont.)
The distribution of is completely characterized under even though
is unknown !!!
Thus, we can set a threshold in the test:
in order to obtain
CFAR !
Furthermore, as the likelihood ratio for non-central t is monotone, this
test is UMPI for testing in the distribution
when is unknown !

42. CFAR (cont.)
The actual probability of detection depends on the actual value of the SNR

43. Summary
Basic concepts
Neyman-Pearson lemma
UMP
Invariance
CFAR