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## Hypothesis Testing

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Agenda

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

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:

vs.

M-ary test:

vs.

vs.

Basic conceptsthen

is said to be a simple hypothesis

vs.

simple vs. composite hypotheses

Example:

A two-sided test: the alternative lies on both sides of

vs.

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 ?

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.)composite:

i.e. the worst case

simple:

composite:

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

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

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

Basic concepts (cont.)Receiving Operating Characteristics (ROC):

Chance line

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 lemmaNote 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.)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

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

Binary comm. (cont.)

Assume equal energies: and define

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

UMP TestsThe Neyman-Pearson lemma holds for simple hypotheses.

Uniformly Most Powerful tests generalize to composite hypotheses

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

pair

, then the test:

Is the UMP test of size for testing

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

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

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

UMP Tests (cont.)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 sufficiencyNo 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

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:

UMP Tests (cont.)UMP one-sided tests exist for a host of problems !

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

Source

Mapper

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

Oops

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

Invariance (cont.)Project the measurement on the subspace orthogonal to the disturbance!

Optimal

signals ?

Let G denote a group of transformations.

X has probability distribution:

Invariance (formal discussion)The measurement is distributed as

Invariance (cont.)Revisit the previous example (AWGN channel with unknown bias)

Consider the group of transformations:

The hypothesis test problem is invariant to G

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

Invariance (another example)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)

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

The False-Alarm Rate is Constant. Thus: CFAR

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

Redraw the problem as:

CFAR (cont.)Utilize Invariance !!

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 !

CFAR (cont.)

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

Summary

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

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