1. ECE 7251: Signal Detection and Estimation
Spring 2002
Prof. Aaron Lanterman
Georgia Institute of Technology
Lecture 26, 3/15/01:
Uniformly Most Powerful Tests

2. An Introductory Case Usual parametric data model
Consider a composite problem:
A test is unformly most powerful of level =PFA if it has a better PD (or at least as good as) than any other –level test
Hero uses notation  instead of PD

3. Graphical Interpretation

4. Finding UMP Tests (When They Exist)
Find the most powerful -level (recall =PFA) test for a fixed 
Just the Neyman-Pearson test
If the decision regions do not vary with , then the test if UMP

5. Gaussian Mean Example Suppose we have n i.i.d samples
Assume is known, but is not
Consider three cases
Sufficies to use

6. The Gaussian Likelihood Ratio

7. Case I: >0 Set the threshold to get the right “level”
Notice the test does not depend on ; hence, it is UMP

8. Power of the Single-Sided Test (Case I)

9. Case II: <0
Notice flip!
Case II is UMP also

10. Power of the Single-Sided Test (Case II)

11. Case III: 0
Uh oh… we can’t just absorb  into the threshold anymore without effecting the inequalities!
Decision region varies with sign of 
No UMP test exists

12. Cauchy Median Example Suppose we have a single sample from the density
Likelihood ratio is
Decision region depends on ,
so no UMP exists!

13. The Monotone Likelihood Ratio Condition
Suppose we have a Fisher Factorization
A UMP test of any level  exists if the likelihood ratio is either monotone increasing or decreasing in T for all
(abuse)

14. Densities Satisfying Monotone Likelihood Ratio Condition
Suppose we have a one-sided test: or
The following satisfy the MLR condition:
i.i.d. samples from 1-D exponential family (Gaussian, Bernoulli, Exponential, Poisson, Gamma, Beta)
i.i.d. samples from uniform density U(0,)
i.i.d. samples from shifted Laplace

15. Densities Not Satisfying Monotone Likelihood Ratio Condition
Gaussian with single-sided H1 on mean but unknown variance
Cauchy density with single sided H1 on centrality parameter
Exponential family with double-sided H1