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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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hypothesis testing

Hypothesis Testing


Lihu Berman

  • Basic concepts
  • Neyman-Pearson lemma
  • UMP
  • Invariance
  • CFAR
basic concepts

Xis 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:


M-ary test:



Basic concepts
basic concepts cont



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 ?

basic concepts cont1

Introduce the test function for binary test of hypotheses:

If the measurement is in the Acceptance region – is accepted

If it is in the Rejection region – is rejected, and is accepted.

is a disjoint covering of the measurement space

Basic concepts (cont.)
basic concepts cont2



i.e. the worst case



Basic concepts (cont.)

Probability of False-Alarm (a.k.a. Size):

Detection probability (a.k.a. Power):

basic concepts cont3

The best test of size has the largest among all tests of that size

Basic concepts (cont.)

Receiving Operating Characteristics (ROC):

Chance line

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

The Neyman-Pearson lemma
the neyman pearson cont1

Note 1: If then the most powerful test is:

Note 2: Introduce the likelihood function:

Then the most powerful test can be written as:

The Neyman-Pearson (cont.)
the neyman pearson cont2

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

The Neyman-Pearson (cont.)
binary comm in awgn



‘1’ = Enemy spotted. ‘0’ = All is clear.

Prior probabilities unknown !

Binary comm. in AWGN
binary comm in awgn1
Binary comm. in AWGN

Natural logarithm is monotone, enabling the use of Log-Likelihood

binary comm cont1
Binary comm. (cont.)

Assume equal energies: and define

ump tests

A test is UMP of size , if for any other test , we have:

UMP Tests

The Neyman-Pearson lemma holds for simple hypotheses.

Uniformly Most Powerful tests generalize to composite hypotheses

ump tests cont

Consider scalar R.Vs whose PDFs are parameterized by scalar

Karlin-Rubin Theorem (for UMP one-sided tests):

If the likelihood-ratio is monotone non-decreasing in x for any


, then the test:

Is the UMP test of size for testing

UMP Tests (cont.)
ump tests cont1

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

UMP Tests (cont.)
ump tests cont2

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

UMP Tests (cont.)
ump tests cont3

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

UMP Tests (cont.)
a note on sufficiency

The statistic T(x) is sufficient for if and only if

Fisher-Neyman factorization theorem:

The statistic T(x) is sufficient for if and only if

A 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

ump tests cont4

Theorem: the one-parameter exponential family of distributions with


has a monotone likelihood ratio in the sufficient statistic

provided that is non-decreasing


UMP Tests (cont.)

UMP one-sided tests exist for a host of problems !

ump tests cont6

Therefore, the test

is the Uniformly Most Powerful test of size for testing

UMP Tests (cont.)

Revisit the binary communication example, but with a slight change.



So what?! Let us continue with the log-likelihood as before…


invariance cont

Intuitively: search for a statistic that is invariant to the nuisance parameter

Invariance (cont.)

Project the measurement on the subspace orthogonal to the disturbance!


signals ?

invariance cont1

The measurement is distributed as

Invariance (cont.)

Revisit the previous example (AWGN channel with unknown bias)

invariance another example

Another example

Consider the group of transformations:

The hypothesis test problem is invariant to G

Invariance (another example)
invariance another example1

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

Invariance (another example)
invariance umpi summary
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.
cfar introductory example
CFAR (introductory example)

Project the measurement on the signal space. A UMP test !

The False-Alarm Rate is Constant. Thus: CFAR

cfar cont

m depends now on the unknown. Test is useless. Certainly not CFAR

Redraw the problem as:

CFAR (cont.)

Utilize Invariance !!

cfar cont1

As before:

Change slightly:


CFAR (cont.)
cfar cont2

The distribution of is completely characterized under even though

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.)


cfar cont3
CFAR (cont.)

The actual probability of detection depends on the actual value of the SNR

  • Basic concepts
  • Neyman-Pearson lemma
  • UMP
  • Invariance
  • CFAR