MS_NotesWeek12 - 114 CHAPTER 2 ELEMENTS OF STATISTICAL...

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114 CHAPTER 2. ELEMENTS OF STATISTICAL INFERENCE 2.5 Hypothesis Testing We assume that Y 1 , . . . , Y n have a joint distribution which depends on the un- known parameters ϑ = ( ϑ 1 , . . . , ϑ p ) T . The set of all possible values of ϑ is called the parameter space Θ . Example 2.31 . Parameter spaces for various distributions: 1. Y i iid Bin( m, p ) , where m is known, i = 1 , 2 , . . . , n . Then ϑ = p and Θ = (0 , 1) . 2. Y i iid Poisson( λ ) , i = 1 , 2 , . . . , n . Then ϑ = λ and Θ = R + . 3. Y i iid N ( μ, σ 2 ) , i = 1 , 2 , . . . , n . Then ϑ = ( μ, σ 2 ) T and Θ = R × R + . 4. Y i ∼ N ( β 0 + β 1 x i , σ 2 ) independently for i = 1 , 2 , . . . , n . Then ϑ = ( β 0 , β 1 , σ 2 ) T and Θ = R 2 × R + . A hypothesis restricts ϑ to lie in Θ star , where Θ star Θ . If we wish to test whether ϑ Θ star , then we test the null hypothesis H 0 : ϑ Θ star against the alternative hypothesis H 1 : ϑ Θ \ Θ star . Example 2.32 . Examples of sets Θ star restricted by null hypotheses: 1. Y i iid Poisson( λ ) , i = 1 , 2 , . . . , n . If we test H 0 : λ = λ 0 against H 1 : λ negationslash = λ 0 then Θ star = { λ 0 } . 2. Y i iid N ( μ, σ 2 ) , i = 1 , 2 , . . . , n . If we test H 0 : μ = μ 0 against H 1 : μ negationslash = μ 0 then Θ star = { μ 0 } × R + . 3. Y i ∼ N ( β 0 + β 1 x i , σ 2 ) independently for i = 1 , 2 , . . . , n . For testing H 0 : β 1 = 0 against H 1 : β 1 negationslash = 0 we have Θ star = R × { 0 } × R + . If Θ star is a single point, the hypothesis is simple . Otherwise, the hypothesis is composite . The set of all possible values y of the random sample Y is called the sample space Y .
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2.5. HYPOTHESIS TESTING 115 Example 2.33 . Examples of sample spaces: 1. Y i iid Poisson( λ ) , i = 1 , 2 , . . . , n . Then Y = [ Z + ∪ { 0 } ] n . 2. Y i iid N ( μ, σ 2 ) , i = 1 , 2 , . . . , n . Then Y = R n . The rejection region is the subset R ⊆ Y such that we reject H 0 if y ∈ R . 2.5.1 Type I and II errors There are two errors in hypothesis testing. A type I error is to reject H 0 when H 0 is true and a type II error is to accept H 0 when H 0 is false. When H 0 is simple, that is, we have H 0 : ϑ = ϑ 0 , the significance level or size of the test is defined to be α = P ( type I error ) = P ( Y ∈ R| H 0 true ) = P ( Y ∈ R| ϑ = ϑ 0 ) . When H 0 is composite, that is, we have H 0 : ϑ Θ star , where Θ star is not a single point, then we have a significance function α ( ϑ ) = P ( Y ∈ R| ϑ ) which depends on the particular value in Θ star taken by ϑ . In this case, the size of the test or significance level is defined to be α = max ϑ Θ star α ( ϑ ) . When H 1 is simple, that is, we have H 1 : ϑ = ϑ 1 , the power of the test is defined to be β = 1 - P ( type II error ) = 1 - P ( Y ∈ R c | H 1 true ) = P ( Y ∈ R| H 1 true ) = P ( Y ∈ R| ϑ = ϑ 1 ) , where R c is a complement of R (not a rejection region).
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116 CHAPTER 2. ELEMENTS OF STATISTICAL INFERENCE When H 1 is composite, that is, we have H 1 : ϑ Θ \ Θ star , then we have a power function β ( ϑ ) = P ( Y ∈ R| ϑ ) which depends on the value in Θ \ Θ star taken by ϑ . 2.5.2 Neyman-Pearson lemma Suppose that H 0 and H 1 are both simple hypotheses, so that we wish to test H 0 : ϑ = ϑ 0 against H 1 : ϑ = ϑ 1 . Then we can compare many different tests with the same significance level α . We will choose the one with the largest power, that is, for a fixed α = P ( type I error ) we choose the test that maximizes β = 1 - P ( type II error ) . The following famous result gives a way of doing this.
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