Likelihood Ratio Test Lecture 19 Biostatistics 602 - Statistical - PowerPoint PPT Presentation

• Probability of type II error = . . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang c , which is typically not possible in practice. for all , for all An ideal test should have power function satisfying c . if . . The power function of a hypothesis test with rejection region R is the Definition - The power function . . . . . . . . . Recap LRT Unbiased Test . Summary Power function 4 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . function of θ defined by β ( θ ) = Pr ( X ∈ R | θ ) = Pr ( reject H 0 | θ ) If θ ∈ Ω c 0 (alternative is true), the probability of rejecting H 0 is called the power of test for this particular value of θ . • Probability of type I error = β ( θ ) if θ ∈ Ω 0 .

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang c , which is typically not possible in practice. for all , for all An ideal test should have power function satisfying The power function of a hypothesis test with rejection region R is the . . Definition - The power function . Power function Summary . . . . . . . . Recap LRT 4 / 33 . Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . function of θ defined by β ( θ ) = Pr ( X ∈ R | θ ) = Pr ( reject H 0 | θ ) If θ ∈ Ω c 0 (alternative is true), the probability of rejecting H 0 is called the power of test for this particular value of θ . • Probability of type I error = β ( θ ) if θ ∈ Ω 0 . • Probability of type II error = 1 − β ( θ ) if θ ∈ Ω c 0 .

. Summary March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang The power function of a hypothesis test with rejection region R is the . . Definition - The power function . Power function . . . . . . Unbiased Test 4 / 33 . . . . Recap LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . function of θ defined by β ( θ ) = Pr ( X ∈ R | θ ) = Pr ( reject H 0 | θ ) If θ ∈ Ω c 0 (alternative is true), the probability of rejecting H 0 is called the power of test for this particular value of θ . • Probability of type I error = β ( θ ) if θ ∈ Ω 0 . • Probability of type II error = 1 − β ( θ ) if θ ∈ Ω c 0 . An ideal test should have power function satisfying β ( θ ) = 0 for all θ ∈ Ω 0 , β ( θ ) = 1 for all θ ∈ Ω c 0 , which is typically not possible in practice.

. is a level . . . . . . . . A test with power function test if Level sup In other words, the maximum probability of making a type I error is equal or less than . Any size test is also a level test Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 test . . . . . . . . . . . Recap LRT Unbiased Test . Summary Sizes and Levels of Tests . . . sup In other words, the maximum probability of making a type I error is 5 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . Size α test A test with power function β ( θ ) is a size α test if β ( θ ) = α θ ∈ Ω 0

. is a level . . . . . . . . A test with power function test if Level sup In other words, the maximum probability of making a type I error is equal or less than . Any size test is also a level test Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 test . . Summary . . . . . . . . Recap LRT Unbiased Test . Sizes and Levels of Tests . . . sup 5 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . Size α test A test with power function β ( θ ) is a size α test if β ( θ ) = α θ ∈ Ω 0 In other words, the maximum probability of making a type I error is α .

. or less than . . . . sup In other words, the maximum probability of making a type I error is equal . . Any size test is also a level test Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 sup 5 / 33 . Summary LRT . Unbiased Test . . . . . . . Sizes and Levels of Tests . . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . Size α test A test with power function β ( θ ) is a size α test if β ( θ ) = α θ ∈ Ω 0 In other words, the maximum probability of making a type I error is α . Level α test A test with power function β ( θ ) is a level α test if β ( θ ) ≤ α θ ∈ Ω 0

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang test test is also a level Any size In other words, the maximum probability of making a type I error is equal sup . . . . . sup 5 / 33 Sizes and Levels of Tests . Summary . . . . . . . . Unbiased Test . LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . Size α test A test with power function β ( θ ) is a size α test if β ( θ ) = α θ ∈ Ω 0 In other words, the maximum probability of making a type I error is α . Level α test A test with power function β ( θ ) is a level α test if β ( θ ) ≤ α θ ∈ Ω 0 or less than α .

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang In other words, the maximum probability of making a type I error is equal sup . . . sup . . . 5 / 33 Sizes and Levels of Tests LRT . . . . . . . . Recap Unbiased Test . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . Size α test A test with power function β ( θ ) is a size α test if β ( θ ) = α θ ∈ Ω 0 In other words, the maximum probability of making a type I error is α . Level α test A test with power function β ( θ ) is a level α test if β ( θ ) ≤ α θ ∈ Ω 0 or less than α . Any size α test is also a level α test

. is the MLE of x L x L x where is the MLE of over , and over sup (restricted MLE). The likelihood ratio test is a test that rejects H if and only if x c where c . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 L x . L . . . . . . . . Recap LRT Unbiased Test . Summary Likelihood Ratio Tests (LRT) . Definition . . x sup 6 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . Let L ( θ | x ) be the likelihood function of θ . The likelihood ratio test statistic for testing H 0 : θ ∈ Ω 0 vs. H 1 : θ ∈ Ω c 0 is

. The likelihood ratio test is a test that rejects H if and only if is the MLE of over , and is the MLE of over (restricted MLE). x . c where c . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 where 6 / 33 Likelihood Ratio Tests (LRT) . . Summary LRT . Definition Recap . . . . . . . . . Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . Let L ( θ | x ) be the likelihood function of θ . The likelihood ratio test statistic for testing H 0 : θ ∈ Ω 0 vs. H 1 : θ ∈ Ω c 0 is sup θ ∈ Ω L ( θ | x ) = L (ˆ sup θ ∈ Ω 0 L ( θ | x ) θ 0 | x ) λ ( x ) = L (ˆ θ | x )

. Definition March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang . c where c x The likelihood ratio test is a test that rejects H if and only if (restricted MLE). . . . 6 / 33 . Unbiased Test Summary . . . . . . . . . Recap Likelihood Ratio Tests (LRT) LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . Let L ( θ | x ) be the likelihood function of θ . The likelihood ratio test statistic for testing H 0 : θ ∈ Ω 0 vs. H 1 : θ ∈ Ω c 0 is sup θ ∈ Ω L ( θ | x ) = L (ˆ sup θ ∈ Ω 0 L ( θ | x ) θ 0 | x ) λ ( x ) = L (ˆ θ | x ) where ˆ θ is the MLE of θ over θ ∈ Ω , and ˆ θ 0 is the MLE of θ over θ ∈ Ω 0

. Summary March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang (restricted MLE). . . . Definition . Likelihood Ratio Tests (LRT) 6 / 33 . Unbiased Test LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Let L ( θ | x ) be the likelihood function of θ . The likelihood ratio test statistic for testing H 0 : θ ∈ Ω 0 vs. H 1 : θ ∈ Ω c 0 is sup θ ∈ Ω L ( θ | x ) = L (ˆ sup θ ∈ Ω 0 L ( θ | x ) θ 0 | x ) λ ( x ) = L (ˆ θ | x ) where ˆ θ is the MLE of θ over θ ∈ Ω , and ˆ θ 0 is the MLE of θ over θ ∈ Ω 0 The likelihood ratio test is a test that rejects H 0 if and only if λ ( x ) ≤ c where 0 ≤ c ≤ 1 .

. x i . . . . . L x n i exp n . exp n i x i We need to find MLE of over and . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . . . Solution . . . . . . . . Recap LRT Unbiased Test . Summary Example of LRT . Problem . . i.i.d. H H For the LRT test and its power function . 7 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n

. x i . . . . . . L x n i exp n . exp n i x i We need to find MLE of over and . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . Solution . . . . . . . . . . Recap LRT Unbiased Test . Summary Example of LRT . Problem . . i.i.d. H For the LRT test and its power function 7 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ ≤ θ 0 H 0

. exp . . . . . . . L x n i x i Solution n exp n i x i We need to find MLE of over and . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . . . For the LRT test and its power function . . . . . . . . Recap LRT Unbiased Test . Summary Example of LRT . Problem . . i.i.d. 7 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ ≤ θ 0 H 0 : θ > θ 0 H 1

. i Solution . . n n exp n x i For the LRT test and its power function We need to find MLE of over and . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . . i.i.d. Example of LRT . . . . . . . . Recap LRT Unbiased Test . Summary 7 / 33 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ ≤ θ 0 H 0 : θ > θ 0 H 1 − ( x i − θ ) 2 1 [ ] ∏ √ L ( θ | x ) = 2 σ 2 2 πσ 2 exp i =1

. exp . Solution . . . n We need to find MLE of i.i.d. over and . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 For the LRT test and its power function 7 / 33 . LRT Recap . . . . . . . . Unbiased Test . Summary Example of LRT . Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ ≤ θ 0 H 0 : θ > θ 0 H 1 − ( x i − θ ) 2 1 [ ] ∏ √ L ( θ | x ) = 2 σ 2 2 πσ 2 exp i =1 i =1 ( x i − θ ) 2 ( 1 ) n [ ∑ n ] √ − = 2 σ 2 2 πσ 2

. Problem March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang exp n . . Solution . For the LRT test and its power function i.i.d. . . . . Unbiased Test . . . . Example of LRT . . . . Recap LRT 7 / 33 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ ≤ θ 0 H 0 : θ > θ 0 H 1 − ( x i − θ ) 2 1 [ ] ∏ √ L ( θ | x ) = 2 σ 2 2 πσ 2 exp i =1 i =1 ( x i − θ ) 2 ( 1 ) n [ ∑ n ] √ − = 2 σ 2 2 πσ 2 We need to find MLE of θ over Ω = ( −∞ , ∞ ) and Ω 0 = ( −∞ , θ 0 ] .

. n x i n i x i x i n n i x i i n x i The equation above minimizes when n i x i n x . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 i 8 / 33 . Summary . . . . . . . . Recap LRT Unbiased Test . , or . . . . . . . . . . . . . . . . . . . . . . . . . . . MLE of θ over Ω = ( −∞ , ∞ ) i =1 ( x i − θ ) 2 [ ∑ n ] To maximize L ( θ | x ) , we need to maximize exp − 2 σ 2 i =1 ( x i − θ ) 2 . equivalently to minimize ∑ n

. x i . n n n i x i n i The equation above minimizes when , or n i x i n x . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 n 8 / 33 . . . . . . . . . Summary Recap LRT Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . MLE of θ over Ω = ( −∞ , ∞ ) i =1 ( x i − θ ) 2 [ ∑ n ] To maximize L ( θ | x ) , we need to maximize exp − 2 σ 2 i =1 ( x i − θ ) 2 . equivalently to minimize ∑ n i + θ 2 − 2 θ x i ) ∑ ( x i − θ ) 2 ∑ ( x 2 = i =1 i =1

. n . n n n i The equation above minimizes when i , or x i n x . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 n 8 / 33 Summary . . . . . . . . Recap LRT Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . MLE of θ over Ω = ( −∞ , ∞ ) i =1 ( x i − θ ) 2 [ ∑ n ] To maximize L ( θ | x ) , we need to maximize exp − 2 σ 2 i =1 ( x i − θ ) 2 . equivalently to minimize ∑ n i + θ 2 − 2 θ x i ) ∑ ( x i − θ ) 2 ∑ ( x 2 = i =1 i =1 n θ 2 − 2 θ ∑ ∑ x 2 = x i + i =1 i =1

. , or March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang n i n n n . n 8 / 33 Summary Unbiased Test Recap . . . . . LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MLE of θ over Ω = ( −∞ , ∞ ) i =1 ( x i − θ ) 2 [ ∑ n ] To maximize L ( θ | x ) , we need to maximize exp − 2 σ 2 i =1 ( x i − θ ) 2 . equivalently to minimize ∑ n i + θ 2 − 2 θ x i ) ∑ ( x i − θ ) 2 ∑ ( x 2 = i =1 i =1 n θ 2 − 2 θ ∑ ∑ x 2 = x i + i =1 i =1 ∑ n The equation above minimizes when θ = ˆ θ = i =1 x i = x .

• However, if x . . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang if X if X X To summarize, . Therefore , the likelihood function will be an increasing function. , and , x does not fall into a valid range of n 9 / 33 LRT . . Summary . Unbiased Test . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MLE of θ over Ω 0 = ( −∞ , θ 0 ] ∑ n • L ( θ | x ) is maximized at θ = = x if x ≤ θ 0 . i =1 x i

. Summary March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang if X if X X To summarize, n . 9 / 33 . . . LRT . . . . . . Unbiased Test Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . MLE of θ over Ω 0 = ( −∞ , θ 0 ] ∑ n • L ( θ | x ) is maximized at θ = = x if x ≤ θ 0 . i =1 x i • However, if x ≥ θ 0 , x does not fall into a valid range of ˆ θ 0 , and θ ≤ θ 0 , the likelihood function will be an increasing function. Therefore ˆ θ 0 = θ 0 .

. Unbiased Test March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang To summarize, . Summary . n 9 / 33 LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MLE of θ over Ω 0 = ( −∞ , θ 0 ] ∑ n • L ( θ | x ) is maximized at θ = = x if x ≤ θ 0 . i =1 x i • However, if x ≥ θ 0 , x does not fall into a valid range of ˆ θ 0 , and θ ≤ θ 0 , the likelihood function will be an increasing function. Therefore ˆ θ 0 = θ 0 . { X if X ≤ θ 0 ˆ θ 0 = θ 0 if X > θ 0

if X n x . . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang . and x c exp Therefore, the likelihood test rejects the null hypothesis if and only if if X n x exp exp exp 10 / 33 . Likelihood ratio test Unbiased Test LRT Recap . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 if X ≤ θ 0  λ ( x ) = L (ˆ i =1( xi − θ 0)2 θ 0 | x )  [ ] ∑ n  − = 2 σ 2 L (ˆ if X > θ 0 θ | x ) i =1( xi − x )2 [ ]  ∑ n  −  2 σ 2

. exp March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang . and x c n x exp Therefore, the likelihood test rejects the null hypothesis if and only if exp exp . 10 / 33 Unbiased Test Likelihood ratio test . . . . Summary LRT . . . . . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 if X ≤ θ 0  λ ( x ) = L (ˆ i =1( xi − θ 0)2 θ 0 | x )  [ ] ∑ n  − = 2 σ 2 L (ˆ if X > θ 0 θ | x ) i =1( xi − x )2 [ ]  ∑ n  −  2 σ 2 { 1 if X ≤ θ 0 = [ − n ( x − θ 0 ) 2 ] if X > θ 0 2 σ 2

. Summary March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang exp Therefore, the likelihood test rejects the null hypothesis if and only if exp exp . exp Likelihood ratio test 10 / 33 . . . . . . . . . Recap Unbiased Test LRT . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 if X ≤ θ 0  λ ( x ) = L (ˆ i =1( xi − θ 0)2 θ 0 | x )  [ ] ∑ n  − = 2 σ 2 L (ˆ if X > θ 0 θ | x ) i =1( xi − x )2 [ ]  ∑ n  −  2 σ 2 { 1 if X ≤ θ 0 = [ − n ( x − θ 0 ) 2 ] if X > θ 0 2 σ 2 − n ( x − θ 0 ) 2 [ ] ≤ c 2 σ 2 and x ≥ θ 0 .

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang x n log c x n log c x log c n x c 11 / 33 . . . . Specifying c Summary . Unbiased Test exp LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − n ( x − θ 0 ) 2 [ ] ≤ 2 σ 2

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang x n log c x n log c x log c c 11 / 33 exp Unbiased Test . . . . . . . . Recap LRT . Summary Specifying c . . . . . . . . . . . . . . . . . . . . . . . . . . . − n ( x − θ 0 ) 2 [ ] ≤ 2 σ 2 ⇒ − n ( x − θ 0 ) 2 ⇐ ≤ 2 σ 2

. exp March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang x n log c x n log c c . 11 / 33 Specifying c LRT . . Unbiased Test . . . . . . . Summary Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . − n ( x − θ 0 ) 2 [ ] ≤ 2 σ 2 ⇒ − n ( x − θ 0 ) 2 ⇐ ≤ 2 σ 2 − 2 σ 2 log c ⇒ ( x − θ 0 ) 2 ⇐ ≥

. Summary March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang n n log c . exp Specifying c c . Recap . . . . . . . . Unbiased Test 11 / 33 LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . − n ( x − θ 0 ) 2 [ ] ≤ 2 σ 2 ⇒ − n ( x − θ 0 ) 2 ⇐ ≤ 2 σ 2 − 2 σ 2 log c ⇒ ( x − θ 0 ) 2 ⇐ ≥ √ − 2 σ 2 log c ⇐ ⇒ x − θ 0 ≥ ( ∵ x > θ 0 )

. Specifying c (cont’d) March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang c n x x Therefore, the rejection region is n n . 12 / 33 Summary LRT . . . Unbiased Test . . . . . . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . So, LRT rejects H 0 if and only if √ − 2 σ 2 log c x − θ 0 ≥ √ − 2 σ 2 log c ⇒ x − θ 0 = c ∗ ⇐ σ / √ n ≥ σ / √ n

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang Therefore, the rejection region is n n . Specifying c (cont’d) Summary 12 / 33 Unbiased Test LRT . . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . So, LRT rejects H 0 if and only if √ − 2 σ 2 log c x − θ 0 ≥ √ − 2 σ 2 log c ⇒ x − θ 0 = c ∗ ⇐ σ / √ n ≥ σ / √ n { x : x − θ 0 } σ / √ n ≥ c ∗

. X n n c Since X X n i.i.d. , X n . Therefore, n Pr = Pr Z n c where Z . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 X c . n . . . . . . . . Recap LRT Unbiased Test . Summary Power function Pr X 13 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . ( X − θ 0 ) σ / √ n ≥ c ∗ β ( θ ) = Pr ( reject H 0 ) = Pr

. X n n c Since X X n i.i.d. , X n . Therefore, n Pr = Pr Z n c where Z . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 X 13 / 33 . Power function . . . . . . . . Recap LRT Unbiased Test . Summary Pr . . . . . . . . . . . . . . . . . . . . . . . . . . . ( X − θ 0 ) σ / √ n ≥ c ∗ β ( θ ) = Pr ( reject H 0 ) = Pr ( X − θ + θ − θ 0 ) ≥ c ∗ σ / √ n =

. = Since X X n i.i.d. , X n . Therefore, X n Pr . Z n c where Z . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 Pr 13 / 33 Pr . . . . . Unbiased Test LRT Summary Recap . . . . Power function . . . . . . . . . . . . . . . . . . . . . . . . . . . ( X − θ 0 ) σ / √ n ≥ c ∗ β ( θ ) = Pr ( reject H 0 ) = Pr ( X − θ + θ − θ 0 ) ≥ c ∗ σ / √ n = ( X − θ σ / √ n ≥ θ 0 − θ ) σ / √ n + c ∗ =

. Pr Pr i.i.d. n . Therefore, X n = Z Pr n c where Z . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . 13 / 33 Recap . . . . LRT Unbiased Test . Summary Power function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( X − θ 0 ) σ / √ n ≥ c ∗ β ( θ ) = Pr ( reject H 0 ) = Pr ( X − θ + θ − θ 0 ) ≥ c ∗ σ / √ n = ( X − θ σ / √ n ≥ θ 0 − θ ) σ / √ n + c ∗ = ( ) θ, σ 2 ∼ N ( θ, σ 2 ) , X ∼ N Since X 1 , · · · , X n

. Power function March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang Pr = . Therefore, n i.i.d. . Pr Pr 13 / 33 Summary Recap LRT . . . . . . . . . Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . ( X − θ 0 ) σ / √ n ≥ c ∗ β ( θ ) = Pr ( reject H 0 ) = Pr ( X − θ + θ − θ 0 ) ≥ c ∗ σ / √ n = ( X − θ σ / √ n ≥ θ 0 − θ ) σ / √ n + c ∗ = ( ) θ, σ 2 ∼ N ( θ, σ 2 ) , X ∼ N Since X 1 , · · · , X n X − θ σ / √ n ∼ N (0 , 1) ( Z ≥ θ 0 − θ ) σ / √ n + c ∗ ⇒ β ( θ ) = where Z ∼ N (0 , 1) .

LRT test rejects H if and only if x . is maximized when c c z Note that Pr Z n c ). is maximum (i.e. c Therefore, size n z . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 Pr Z n . LRT . . . . . . . . Recap 14 / 33 Z Unbiased Test . Summary sup sup Pr . . . . . . . . . . . . . . . . . . . . . . . . . . . Making size α LRT To make a size α test,

LRT test rejects H if and only if x . c c Pr Z c c z Note that Pr Z n is maximum (i.e. is maximized when Z ). Therefore, size n z . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 n sup Pr . LRT . . . . . . . . Recap Unbiased Test . Summary sup 14 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . Making size α LRT To make a size α test, β ( θ ) = α θ ∈ Ω 0

LRT test rejects H if and only if x . is maximized when Pr Z c c z Note that Pr Z n c is maximum (i.e. Pr ). Therefore, size n z . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . 14 / 33 sup Summary . . . . . . . . Recap LRT sup Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . Making size α LRT To make a size α test, β ( θ ) = α θ ∈ Ω 0 ( Z ≥ θ 0 − θ ) σ / √ n + c ∗ = α θ ≤ θ 0

LRT test rejects H if and only if x . is maximized when . c z Note that Pr Z n c ). is maximum (i.e. sup Therefore, size n z . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 Pr 14 / 33 Summary . . . . sup Unbiased Test LRT . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Making size α LRT To make a size α test, β ( θ ) = α θ ∈ Ω 0 ( Z ≥ θ 0 − θ ) σ / √ n + c ∗ = α θ ≤ θ 0 Pr ( Z ≥ c ∗ ) = α

LRT test rejects H if and only if x . sup March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang z . n Therefore, size Note that Pr . Pr sup 14 / 33 Unbiased Test . Summary Recap . . . . . . . LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . Making size α LRT To make a size α test, β ( θ ) = α θ ∈ Ω 0 ( Z ≥ θ 0 − θ ) σ / √ n + c ∗ = α θ ≤ θ 0 Pr ( Z ≥ c ∗ ) = α c ∗ = z α ( σ / √ n + c ∗ ) Z ≥ θ 0 − θ is maximized when θ is maximum (i.e. θ = θ 0 ).

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang Note that Pr . Pr sup sup Summary 14 / 33 Unbiased Test LRT . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Making size α LRT To make a size α test, β ( θ ) = α θ ∈ Ω 0 ( Z ≥ θ 0 − θ ) σ / √ n + c ∗ = α θ ≤ θ 0 Pr ( Z ≥ c ∗ ) = α c ∗ = z α ( σ / √ n + c ∗ ) Z ≥ θ 0 − θ is maximized when θ is maximum (i.e. θ = θ 0 ). Therefore, size α LRT test rejects H 0 if and only if x − θ 0 σ / √ n ≥ z α .

n I . . . . L x n i e x i I x i e x i x . The likelihood function is a increasing function of , bounded by x . Therefore, when , L x is maximized when x . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . . . . . . . . . . . . Recap LRT Unbiased Test . Summary Another Example of LRT . Problem . . i.i.d. LRT testing the following one-sided hypothesis. H H . Solution . 15 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ f ( x | θ ) = e − ( x − θ ) where x ≥ θ and −∞ < θ < ∞ . Find a X 1 , · · · , X n

n I x i . . . . . L x n i e x i I x i e . . x The likelihood function is a increasing function of , bounded by x . Therefore, when , L x is maximized when x . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . . . Solution . . . . . . . . Recap LRT Unbiased Test . Summary Another Example of LRT . Problem . . i.i.d. LRT testing the following one-sided hypothesis. . 15 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ f ( x | θ ) = e − ( x − θ ) where x ≥ θ and −∞ < θ < ∞ . Find a X 1 , · · · , X n : θ ≤ θ 0 H 0 : θ > θ 0 H 1

. x . Solution . . n The likelihood function is a increasing function of , bounded by . LRT testing the following one-sided hypothesis. Therefore, when , L x is maximized when x . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . 15 / 33 Unbiased Test . . Summary LRT Recap i.i.d. . . . . Another Example of LRT . . Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ f ( x | θ ) = e − ( x − θ ) where x ≥ θ and −∞ < θ < ∞ . Find a X 1 , · · · , X n : θ ≤ θ 0 H 0 : θ > θ 0 H 1 ∏ e − ( x i − θ ) I ( x i ≥ θ ) L ( θ | x ) = i =1 e − ∑ x i + n θ I ( θ ≤ x (1) ) =

. Therefore, when . . Solution . . n , L i.i.d. x is maximized when x . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 LRT testing the following one-sided hypothesis. 15 / 33 . LRT Recap . . . . Unbiased Test . Summary Another Example of LRT . . . . . Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ f ( x | θ ) = e − ( x − θ ) where x ≥ θ and −∞ < θ < ∞ . Find a X 1 , · · · , X n : θ ≤ θ 0 H 0 : θ > θ 0 H 1 ∏ e − ( x i − θ ) I ( x i ≥ θ ) L ( θ | x ) = i =1 e − ∑ x i + n θ I ( θ ≤ x (1) ) = The likelihood function is a increasing function of θ , bounded by θ ≤ x (1) .

. Problem March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang n . . Solution . . LRT testing the following one-sided hypothesis. i.i.d. . . 15 / 33 . LRT . . . . Another Example of LRT Recap . . . . . Summary Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ f ( x | θ ) = e − ( x − θ ) where x ≥ θ and −∞ < θ < ∞ . Find a X 1 , · · · , X n : θ ≤ θ 0 H 0 : θ > θ 0 H 1 ∏ e − ( x i − θ ) I ( x i ≥ θ ) L ( θ | x ) = i =1 e − ∑ x i + n θ I ( θ ≤ x (1) ) = The likelihood function is a increasing function of θ , bounded by θ ≤ x (1) . Therefore, when θ ∈ Ω = R , L ( θ | x ) is maximized when θ = ˆ θ = x (1) .

. x n e xi nx if x if e n e x if x if x Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 xi x . The likelihood ratio test statistic is . . . . . . . . Recap LRT Unbiased Test . Summary Solution (cont’d) 16 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . When θ ∈ Ω c 0 , the likelihood is still an increasing function, but bounded by θ ≤ min ( x (1) , θ 0 ) . Therefore, the likelihood is maximized when θ = ˆ θ 0 = min ( x (1) , θ 0 ) .

. Solution (cont’d) March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang x if x if x e n . 16 / 33 Summary LRT Unbiased Test . . . . . . . . . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . When θ ∈ Ω c 0 , the likelihood is still an increasing function, but bounded by θ ≤ min ( x (1) , θ 0 ) . Therefore, the likelihood is maximized when θ = ˆ θ 0 = min ( x (1) , θ 0 ) . The likelihood ratio test statistic is { e − ∑ xi + n θ 0 if θ 0 < x (1) e − ∑ xi + nx (1) λ ( x ) = 1 if θ 0 ≥ x (1)

. Unbiased Test March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang . Solution (cont’d) Summary . 16 / 33 LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . When θ ∈ Ω c 0 , the likelihood is still an increasing function, but bounded by θ ≤ min ( x (1) , θ 0 ) . Therefore, the likelihood is maximized when θ = ˆ θ 0 = min ( x (1) , θ 0 ) . The likelihood ratio test statistic is { e − ∑ xi + n θ 0 if θ 0 < x (1) e − ∑ xi + nx (1) λ ( x ) = 1 if θ 0 ≥ x (1) e n ( θ 0 − x (1) ) { if θ 0 < x (1) = if θ 0 ≥ x (1) 1

. X log c n So, LRT reject H is x log c n and x . The power function is Pr log c n n X To find size test, we need to find c satisfying the condition sup Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 x log c . x . . . . . . . . Recap LRT Unbiased Test . Summary Solution (cont’d) e n x c and x 17 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . The LRT rejects H 0 if and only if

. X n So, LRT reject H is x log c n and x . The power function is Pr log c . n X To find size test, we need to find c satisfying the condition sup Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 n log c 17 / 33 Solution (cont’d) . . . . . . . . Recap LRT Unbiased Test . Summary c . . . . . . . . . . . . . . . . . . . . . . . . . . . The LRT rejects H 0 if and only if e n ( θ 0 − x (1) ) ≤ ( and θ 0 < x (1) ) ⇐ ⇒ θ 0 − x (1) ≤ ⇐ ⇒ x (1) ≥ θ 0 − log c

. n n n n Pr X log c X . To find size test, we need to find c satisfying the condition sup Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 log c 17 / 33 c . . . . . . . . . Recap LRT Unbiased Test Summary Solution (cont’d) . . . . . . . . . . . . . . . . . . . . . . . . . . . The LRT rejects H 0 if and only if e n ( θ 0 − x (1) ) ≤ ( and θ 0 < x (1) ) ⇐ ⇒ θ 0 − x (1) ≤ ⇐ ⇒ x (1) ≥ θ 0 − log c So, LRT reject H 0 is x (1) ≥ θ 0 − log c and x (1) > θ 0 . The power function is

. c March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang sup test, we need to find c satisfying the condition To find size n Pr n n . log c n 17 / 33 Unbiased Test . Solution (cont’d) . . . . . . . Summary . Recap LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . The LRT rejects H 0 if and only if e n ( θ 0 − x (1) ) ≤ ( and θ 0 < x (1) ) ⇐ ⇒ θ 0 − x (1) ≤ ⇐ ⇒ x (1) ≥ θ 0 − log c So, LRT reject H 0 is x (1) ≥ θ 0 − log c and x (1) > θ 0 . The power function is ( ) β ( θ ) = X (1) ≤ θ 0 − log c ∧ X (1) > θ 0

. Solution (cont’d) March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang sup n Pr n n . log c c n Summary LRT . . . . . . . . Recap 17 / 33 Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . The LRT rejects H 0 if and only if e n ( θ 0 − x (1) ) ≤ ( and θ 0 < x (1) ) ⇐ ⇒ θ 0 − x (1) ≤ ⇐ ⇒ x (1) ≥ θ 0 − log c So, LRT reject H 0 is x (1) ≥ θ 0 − log c and x (1) > θ 0 . The power function is ( ) β ( θ ) = X (1) ≤ θ 0 − log c ∧ X (1) > θ 0 To find size α test, we need to find c satisfying the condition β ( θ ) = α θ ≤ θ 0

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang for every x in the sample space. x T x . . Theorem 8.2.4 . LRT based on sufficient statistics Summary . Unbiased Test LRT Recap . . . . . . . . 18 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . If T ( X ) is a sufficient statistic for θ , λ ∗ ( t ) is the LRT statistic based on T , and λ ( x ) is the LRT statistic based on x then

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang for every x in the sample space. . . Theorem 8.2.4 . LRT based on sufficient statistics Summary . Unbiased Test LRT Recap . . . . . . . . 18 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . If T ( X ) is a sufficient statistic for θ , λ ∗ ( t ) is the LRT statistic based on T , and λ ( x ) is the LRT statistic based on x then λ ∗ [ T ( x )] = λ ( x )

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang for every x in the sample space. . . Theorem 8.2.4 . LRT based on sufficient statistics Summary . Unbiased Test LRT Recap . . . . . . . . 18 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . If T ( X ) is a sufficient statistic for θ , λ ∗ ( t ) is the LRT statistic based on T , and λ ( x ) is the LRT statistic based on x then λ ∗ [ T ( x )] = λ ( x )

. T x sup L T x t sup L t Then, the LRT statistic based on T X is defined as sup g t sup g t Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 t to be the pdf or pmf of T x . . and we can choose g t . . . . . . . . Recap LRT Unbiased Test . Summary Proof By Factorization Theorem, the joint pdf of x can be written as 19 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . f ( x | θ ) = g ( T ( x ) | θ ) h ( x )

. t L T x t sup L T x sup t g t sup g t Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 sup Then, the LRT statistic based on T X is defined as . Recap . . . . . . . . LRT Unbiased Test . Summary Proof By Factorization Theorem, the joint pdf of x can be written as 19 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . f ( x | θ ) = g ( T ( x ) | θ ) h ( x ) and we can choose g ( t | θ ) to be the pdf or pmf of T ( x ) .

. Unbiased Test March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang By Factorization Theorem, the joint pdf of x can be written as . Summary . Proof 19 / 33 LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f ( x | θ ) = g ( T ( x ) | θ ) h ( x ) and we can choose g ( t | θ ) to be the pdf or pmf of T ( x ) . Then, the LRT statistic based on T ( X ) is defined as sup θ ∈ Ω 0 L ( θ | T ( x ) = t ) sup θ ∈ Ω L ( θ | T ( x ) = t ) = sup θ ∈ Ω 0 g ( t | θ ) λ ∗ ( t ) = sup θ ∈ Ω g ( t | θ )

. g T x f x sup g T x h x sup g T x h x sup sup f x g T x T x The simplified expression of x should depend on x only through T x , where T x is a sufficient statistic for . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 sup sup . x . . . . . . . . Recap LRT Unbiased Test . Summary Proof (cont’d) LRT statistic based on X is x sup L x sup L 20 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . .

. g T x h x sup g T x h x sup g T x sup T x sup The simplified expression of x should depend on x only through T x , where T x is a sufficient statistic for . Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 g T x 20 / 33 . . . . . . . . LRT statistic based on X is Proof (cont’d) Summary . Unbiased Test . LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . sup θ ∈ Ω 0 L ( θ | x ) sup θ ∈ Ω L ( θ | x ) = sup θ ∈ Ω 0 f ( x | θ ) λ ( x ) = sup θ ∈ Ω f ( x | θ )

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang . where T x is a sufficient statistic for x should depend on x only through T x , The simplified expression of T x g T x sup g T x sup 20 / 33 LRT statistic based on X is . . . . . . . . . Recap LRT Unbiased Test Summary Proof (cont’d) . . . . . . . . . . . . . . . . . . . . . . . . . . . sup θ ∈ Ω 0 L ( θ | x ) sup θ ∈ Ω L ( θ | x ) = sup θ ∈ Ω 0 f ( x | θ ) λ ( x ) = sup θ ∈ Ω f ( x | θ ) sup θ ∈ Ω 0 g ( T ( x ) | θ ) h ( x ) = sup θ ∈ Ω g ( T ( x ) | θ ) h ( x )

. Summary March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang . where T x is a sufficient statistic for x should depend on x only through T x , The simplified expression of . LRT statistic based on X is Proof (cont’d) 20 / 33 . LRT . . . . Unbiased Test . . . . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . sup θ ∈ Ω 0 L ( θ | x ) sup θ ∈ Ω L ( θ | x ) = sup θ ∈ Ω 0 f ( x | θ ) λ ( x ) = sup θ ∈ Ω f ( x | θ ) sup θ ∈ Ω 0 g ( T ( x ) | θ ) h ( x ) = sup θ ∈ Ω g ( T ( x ) | θ ) h ( x ) sup θ ∈ Ω 0 g ( T ( x ) | θ ) sup θ ∈ Ω g ( T ( x ) | θ ) = λ ∗ ( T ( x )) =

. Unbiased Test March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang . LRT statistic based on X is Proof (cont’d) Summary . 20 / 33 LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sup θ ∈ Ω 0 L ( θ | x ) sup θ ∈ Ω L ( θ | x ) = sup θ ∈ Ω 0 f ( x | θ ) λ ( x ) = sup θ ∈ Ω f ( x | θ ) sup θ ∈ Ω 0 g ( T ( x ) | θ ) h ( x ) = sup θ ∈ Ω g ( T ( x ) | θ ) h ( x ) sup θ ∈ Ω 0 g ( T ( x ) | θ ) sup θ ∈ Ω g ( T ( x ) | θ ) = λ ∗ ( T ( x )) = The simplified expression of λ ( x ) should depend on x only through T ( x ) , where T ( x ) is a sufficient statistic for θ .

n exp n exp . L . . . . T X X is a sufficient statistic for . T n t sup sup t . L t t n sup t n Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . . . . . . . . . . . . Recap LRT Unbiased Test . Summary Example . Problem . . i.i.d. H H Find a size LRT. . Solution - Using sufficient statistics 21 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n

n exp n exp . sup . . . . . T X X is a sufficient statistic for . T n t t L . sup L t t n sup t n Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . . . Solution - Using sufficient statistics . . . . . . . . Recap LRT Unbiased Test . Summary Example . Problem . . i.i.d. . 21 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ = θ 0 H 0 : θ ̸ = θ 0 H 1 Find a size α LRT.

n exp n exp . sup Solution - Using sufficient statistics . . T n t sup L t t L . t n sup t n Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 . 21 / 33 Example . Recap LRT Unbiased Test . Summary . . Problem . . i.i.d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ = θ 0 H 0 : θ ̸ = θ 0 H 1 Find a size α LRT. T ( X ) = X is a sufficient statistic for θ .

n exp n exp . sup . . T n t sup L t t L . t n sup t n Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 Solution - Using sufficient statistics . 21 / 33 . . . . . . . . . Recap LRT Unbiased Test . Summary Example . Problem . i.i.d. . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ = θ 0 H 0 : θ ̸ = θ 0 H 1 Find a size α LRT. T ( X ) = X is a sufficient statistic for θ . θ, σ 2 ( ) ∼ N

. Problem March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang n T . . Solution - Using sufficient statistics . i.i.d. . . . . LRT . . . . Example . . . . Recap 21 / 33 Summary . Unbiased Test . . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( θ, σ 2 ) where σ 2 is known. Consider X 1 , · · · , X n : θ = θ 0 H 0 : θ ̸ = θ 0 H 1 Find a size α LRT. T ( X ) = X is a sufficient statistic for θ . θ, σ 2 ( ) ∼ N [ − ( t − θ 0 ) 2 ] 1 sup θ ∈ Ω 0 L ( θ | t ) 2 πσ 2 / n exp 2 σ 2 / n λ ( t ) = sup θ ∈ Ω L ( θ | t ) = [ − ( t − θ ) 2 ] 1 sup θ ∈ Ω 2 πσ 2 / n exp 2 σ 2 / n

. = n t LRT rejects H if and only if t exp n t c t t n log c c Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 exp the LRT statistic is . Recap . . . . . . . . LRT Unbiased Test . Summary Solution (cont’d) 22 / 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . The numerator is fixed, and MLE in the denominator is ˆ θ = t . Therefore

. the LRT statistic is March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang c log c n t = c n t exp t LRT rejects H if and only if . 22 / 33 . . . . . . . . Solution (cont’d) Summary . Unbiased Test LRT Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . The numerator is fixed, and MLE in the denominator is ˆ θ = t . Therefore − n ( t − θ 0 ) 2 [ ] λ ( t ) = exp 2 σ 2

. the LRT statistic is March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang c log c n t = c n t exp t . 22 / 33 Unbiased Test Recap Solution (cont’d) . . . . . . . . Summary . LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . The numerator is fixed, and MLE in the denominator is ˆ θ = t . Therefore − n ( t − θ 0 ) 2 [ ] λ ( t ) = exp 2 σ 2 LRT rejects H 0 if and only if

. Solution (cont’d) March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang c log c n t = c . the LRT statistic is 22 / 33 Summary . . . Unbiased Test . . . . . . Recap LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . The numerator is fixed, and MLE in the denominator is ˆ θ = t . Therefore − n ( t − θ 0 ) 2 [ ] λ ( t ) = exp 2 σ 2 LRT rejects H 0 if and only if − n ( t − θ 0 ) 2 [ ] λ ( t ) = exp ≤ 2 σ 2

. . March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang = c . the LRT statistic is Solution (cont’d) Summary 22 / 33 Unbiased Test Recap . . . . . . . . LRT . . . . . . . . . . . . . . . . . . . . . . . . . . . The numerator is fixed, and MLE in the denominator is ˆ θ = t . Therefore − n ( t − θ 0 ) 2 [ ] λ ( t ) = exp 2 σ 2 LRT rejects H 0 if and only if − n ( t − θ 0 ) 2 [ ] λ ( t ) = exp ≤ 2 σ 2 � � t − θ 0 √ − 2 log c = c ∗ � � ⇒ σ / √ n � ≥ � � �

. c sup Pr T n c Pr T n c Pr Z Pr Z A size c Pr Z c Z T n z Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 test satisfies 23 / 33 . Solution (cont’d) . . . . . . . . Recap LRT Unbiased Test . Summary Note that n . . . . . . . . . . . . . . . . . . . . . . . . . . . θ, σ 2 ( ) T = X ∼ N T − θ 0 σ / √ n ∼ N (0 , 1)

. Pr Z . Pr T n c Pr Z c c sup Pr Z c Z T n z Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 Pr 23 / 33 Note that . Summary . Unbiased Test LRT Solution (cont’d) . . . . . Recap . n . . . . . . . . . . . . . . . . . . . . . . . . . . . . θ, σ 2 ( ) T = X ∼ N T − θ 0 σ / √ n ∼ N (0 , 1) A size α test satisfies (� T − θ � ) � ≥ c ∗ � σ / √ n � = α � � θ ∈ Ω 0 �

. c Pr . Pr Pr Z c Pr Z Pr Z n c Z T n z Hyun Min Kang Biostatistics 602 - Lecture 19 March 26th, 2013 sup 23 / 33 Solution (cont’d) Note that Summary . Unbiased Test LRT . Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . θ, σ 2 ( ) T = X ∼ N T − θ 0 σ / √ n ∼ N (0 , 1) A size α test satisfies (� T − θ � ) � ≥ c ∗ � σ / √ n � = α � � θ ∈ Ω 0 � (� � T − θ 0 ) � ≥ c ∗ � � σ / √ n = α � � �