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Uniformly Most Powerful Tests

Uniformly Most Powerful Tests. ECE 7251: Spring 2004 Lecture 27 3/22/04. Prof. Aaron D. Lanterman School of Electrical & Computer Engineering Georgia Institute of Technology AL: 404-385-2548 <lanterma@ece.gatech.edu>. An Introductory Case. Usual parametric data model

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Uniformly Most Powerful Tests

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  1. Uniformly Most Powerful Tests ECE 7251: Spring 2004 Lecture 27 3/22/04 Prof. Aaron D. Lanterman School of Electrical & Computer Engineering Georgia Institute of Technology AL: 404-385-2548 <lanterma@ece.gatech.edu>

  2. An Introductory Case • Usual parametric data model • Consider a composite problem: • A test is unformly most powerfulof 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 is UMP

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

  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. Notice flip! Case II: <0 • 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 havea single sample from the density and we want to decide • 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 MLR 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 Also works for

  15. Densities Not Satisfying MLR Condition • Gaussian with single-sided H1 on mean but unknown variance • Cauchy density with single-sided H1on centrality parameter • Exponential family with double-sided H1

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