. Summary April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang April 2nd, 2013 Hyun Min Kang Wald Test Asymptotics of LRT Lecture 21 Biostatistics 602 - Statistical Inference . . . . Wald Test Asymptotics of LRT Karlin-Rabin Recap . . . . . . . 1 / 25 . . . . . . . . . . . . . . . . . . . .
• What is a Uniformly Most Powerful (UMP) Test? • Does UMP level • For composite hypothesis, which property makes it possible to • What is a sufficient condition for an exponential family to have MLR • For one-sided composite hypothesis testing, if a sufficient statistic . April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang test be constructed? satisfies MLR property, how can a UMP level property? test? construct a UMP level test always exist for simple hypothesis testing? Last Lecture . Summary . Wald Test Asymptotics of LRT Karlin-Rabin Recap . . . . . . . 2 / 25 . . . . . . . . . . . . . . . . . . . .
• Does UMP level • For composite hypothesis, which property makes it possible to • What is a sufficient condition for an exponential family to have MLR • For one-sided composite hypothesis testing, if a sufficient statistic . . April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang test be constructed? satisfies MLR property, how can a UMP level property? test? construct a UMP level test always exist for simple hypothesis testing? Last Lecture Summary . Wald Test Asymptotics of LRT Karlin-Rabin Recap . . . . . . . 2 / 25 . . . . . . . . . . . . . . . . . . . . • What is a Uniformly Most Powerful (UMP) Test?
• For composite hypothesis, which property makes it possible to • What is a sufficient condition for an exponential family to have MLR • For one-sided composite hypothesis testing, if a sufficient statistic . . April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang test be constructed? satisfies MLR property, how can a UMP level property? test? construct a UMP level Last Lecture Summary . Wald Test Asymptotics of LRT Karlin-Rabin Recap . . . . . . . 2 / 25 . . . . . . . . . . . . . . . . . . . . • What is a Uniformly Most Powerful (UMP) Test? • Does UMP level α test always exist for simple hypothesis testing?
• What is a sufficient condition for an exponential family to have MLR • For one-sided composite hypothesis testing, if a sufficient statistic . . April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang test be constructed? satisfies MLR property, how can a UMP level property? Last Lecture . Summary Wald Test Recap . . . . . . . Karlin-Rabin 2 / 25 Asymptotics of LRT . . . . . . . . . . . . . . . . . . . . • What is a Uniformly Most Powerful (UMP) Test? • Does UMP level α test always exist for simple hypothesis testing? • For composite hypothesis, which property makes it possible to construct a UMP level α test?
• For one-sided composite hypothesis testing, if a sufficient statistic . Wald Test April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang test be constructed? satisfies MLR property, how can a UMP level property? Last Lecture . . Summary 2 / 25 . . . Karlin-Rabin . . . . Asymptotics of LRT Recap . . . . . . . . . . . . . . . . . . . . • What is a Uniformly Most Powerful (UMP) Test? • Does UMP level α test always exist for simple hypothesis testing? • For composite hypothesis, which property makes it possible to construct a UMP level α test? • What is a sufficient condition for an exponential family to have MLR
. Asymptotics of LRT April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang property? Last Lecture Summary . Wald Test . 2 / 25 . Karlin-Rabin Recap . . . . . . . . . . . . . . . . . . . . . . . . . . • What is a Uniformly Most Powerful (UMP) Test? • Does UMP level α test always exist for simple hypothesis testing? • For composite hypothesis, which property makes it possible to construct a UMP level α test? • What is a sufficient condition for an exponential family to have MLR • For one-sided composite hypothesis testing, if a sufficient statistic satisfies MLR property, how can a UMP level α test be constructed?
. . April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang in this class. c test has the smallest type II error probability for any UMP level . . . function of another test in C . . . Definition . Uniformly Most Powerful Test (UMP) Summary . . . . . . . Recap Karlin-Rabin 3 / 25 Asymptotics of LRT Wald Test . . . . . . . . . . . . . . . . . . . . . Let C be a class of tests between H 0 : θ ∈ Ω vs H 1 : θ ∈ Ω c 0 . A test in C , with power function β ( θ ) is uniformly most powerful (UMP) test in class C if β ( θ ) ≥ β ′ ( θ ) for every θ ∈ Ω c 0 and every β ′ ( θ ) , which is a power UMP level α test Consider C be the set of all the level α test. The UMP test in this class is called a UMP level α test.
. Uniformly Most Powerful Test (UMP) April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang in this class. . . . function of another test in C . . . Definition . . Summary Karlin-Rabin . . . . . . . Recap . 3 / 25 Asymptotics of LRT Wald Test . . . . . . . . . . . . . . . . . . . . Let C be a class of tests between H 0 : θ ∈ Ω vs H 1 : θ ∈ Ω c 0 . A test in C , with power function β ( θ ) is uniformly most powerful (UMP) test in class C if β ( θ ) ≥ β ′ ( θ ) for every θ ∈ Ω c 0 and every β ′ ( θ ) , which is a power UMP level α test Consider C be the set of all the level α test. The UMP test in this class is called a UMP level α test. UMP level α test has the smallest type II error probability for any θ ∈ Ω c 0
. Uniformly Most Powerful Test (UMP) April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang in this class. . . . function of another test in C . . . Definition . . Summary Karlin-Rabin . . . . . . . Recap . 3 / 25 Asymptotics of LRT Wald Test . . . . . . . . . . . . . . . . . . . . Let C be a class of tests between H 0 : θ ∈ Ω vs H 1 : θ ∈ Ω c 0 . A test in C , with power function β ( θ ) is uniformly most powerful (UMP) test in class C if β ( θ ) ≥ β ′ ( θ ) for every θ ∈ Ω c 0 and every β ′ ( θ ) , which is a power UMP level α test Consider C be the set of all the level α test. The UMP test in this class is called a UMP level α test. UMP level α test has the smallest type II error probability for any θ ∈ Ω c 0
• (Sufficiency) Any test that satisfies 8.3.1 and 8.3.2 is a UMP level • (Necessity) if there exist a test satisfying 8.3.1 and 8.3.2 with k . then every UMP level if f x kf x For some k and Pr X R , Then, test , test is a size x test (satisfies 8.3.2), and every UMP level test satisfies 8.3.1 except perhaps on a set A satisfying Pr X A Pr X A . Hyun Min Kang Biostatistics 602 - Lecture 21 April 2nd, 2013 R c and . Summary . . . . . . . Recap Karlin-Rabin Asymptotics of LRT Wald Test kf x . Neyman-Pearson Lemma . Theorem 8.3.12 - Neyman-Pearson Lemma . . R that satisfies x R if f x 4 / 25 . . . . . . . . . . . . . . . . . . . . Consider testing H 0 : θ = θ 0 vs. H 1 : θ = θ 1 where the pdf or pmf corresponding the θ i is f ( x | θ i ) , i = 0 , 1 , using a test with rejection region
• (Sufficiency) Any test that satisfies 8.3.1 and 8.3.2 is a UMP level • (Necessity) if there exist a test satisfying 8.3.1 and 8.3.2 with k . then every UMP level kf x For some k and Pr X R , Then, test , test is a size R c test (satisfies 8.3.2), and every UMP level test satisfies 8.3.1 except perhaps on a set A satisfying Pr X A Pr X A . Hyun Min Kang Biostatistics 602 - Lecture 21 April 2nd, 2013 if f x x . Summary . . . . . . . Recap Karlin-Rabin Asymptotics of LRT Wald Test . Neyman-Pearson Lemma . Theorem 8.3.12 - Neyman-Pearson Lemma . . R that satisfies 4 / 25 . . . . . . . . . . . . . . . . . . . . Consider testing H 0 : θ = θ 0 vs. H 1 : θ = θ 1 where the pdf or pmf corresponding the θ i is f ( x | θ i ) , i = 0 , 1 , using a test with rejection region x ∈ R if f ( x | θ 1 ) > kf ( x | θ 0 ) (8 . 3 . 1) and
• (Sufficiency) Any test that satisfies 8.3.1 and 8.3.2 is a UMP level • (Necessity) if there exist a test satisfying 8.3.1 and 8.3.2 with k . test is a size and Pr X R , Then, test , then every UMP level test (satisfies 8.3.2), and . every UMP level test satisfies 8.3.1 except perhaps on a set A satisfying Pr X A Pr X A . Hyun Min Kang Biostatistics 602 - Lecture 21 April 2nd, 2013 For some k 4 / 25 . Theorem 8.3.12 - Neyman-Pearson Lemma Recap Karlin-Rabin Asymptotics of LRT Wald Test Summary Neyman-Pearson Lemma . . . . R that satisfies . . . . . . . . . . . . . . . . . . . . . . . . . . Consider testing H 0 : θ = θ 0 vs. H 1 : θ = θ 1 where the pdf or pmf corresponding the θ i is f ( x | θ i ) , i = 0 , 1 , using a test with rejection region x ∈ R if f ( x | θ 1 ) > kf ( x | θ 0 ) (8 . 3 . 1) and x ∈ R c if f ( x | θ 1 ) < kf ( x | θ 0 ) (8 . 3 . 2)
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