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Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com 10.11 Likelihood Ratio Tests 551 Restricting µ to 0 implies that µ = µ 0 , and we can find L( ˆ 0 ) if we determine the value of σ 2 that maximizes L(µ, σ 2 ) subject to the constraint that µ = µ 0 . From Example 9.15, we see that when µ = µ 0 the value of σ 2 that maximizes L(µ 0 ,σ 2 ) is ˆσ 0 2 = 1 n∑ (y i − µ 0 ) 2 . n i=1 Thus, L( ˆ 0 ) is obtained by replacing µ with µ 0 and σ 2 with ˆσ 0 2 in L(µ, σ 2 ), which gives ( ) 1 n ( ) [ ] n/2 n∑ (y i − µ 0 ) L( ˆ 0 ) = √ 2π 1ˆσ 2 ( ) 1 n ( ) n/2 0 2 exp − i=1 2ˆσ 0 2 = √ 2π 1ˆσ 0 2 e −n/2 . We now turn to finding L( ˆ). As in Example 9.15, it is easier to look at ln L(µ, σ 2 ), ln[L(µ, σ 2 )] =− n 2 ln σ 2 − n 1 n∑ ln 2π − (y 2 2σ 2 i − µ) 2 . Taking derivatives with respect to µ and σ 2 , we obtain ∂{ln[L(µ, σ 2 )]} = 1 n∑ (y ∂µ σ 2 i − µ), i=1 ∂{ln[L(µ, σ 2 )]} ( n ) =− + 1 n∑ (y ∂σ 2 2σ 2 2σ 4 i − µ) 2 . We need to find the maximum of L(µ, σ 2 ) over the set ={(µ, σ 2 ) : µ ≥ µ 0 , σ 2 > 0}. Notice that i=1 i=1 ∂ L(µ, σ 2 )/∂µ < 0, if µ>y, ∂ L(µ, σ 2 )/∂µ = 0, if µ = y, ∂ L(µ, σ 2 )/∂µ > 0, if µ 0},lnL(µ, σ 2 ) [and also L(µ, σ 2 )] is maximized at ˆµ where { y, if y >µ0 , ˆµ = µ 0 , if y ≤ µ 0 . Just as earlier, the value of σ 2 in that maximizes L(µ, σ 2 ),is ˆσ 2 = 1 n∑ (y i − ˆµ) 2 . n L( ˆ) is obtained by replacing µ with ˆµ and σ 2 with ˆσ 2 , which yields ( ) 1 n ( ) [ n/2 n∑ L( ˆ) = √ exp − 2π 1ˆσ 2 i=1 i=1 ] (y i − ˆµ) 2 = 2ˆσ 2 ( 1 √ 2π ) n ( 1ˆσ 2 ) n/2 e −n/2 .

552 Chapter 10 Hypothesis Testing www.downloadslide.com Thus, ) n/2 λ = L( ( ˆ 0 ) ˆσ 2 L( ˆ) = ˆσ 0 2 ⎧ [ ∑ n ⎪⎨ i=1 (y ] n/2 i − y) 2 ∑ n = i=1 (y , if y >µ i − µ 0 ) 2 0 ⎪⎩ 1, if y ≤ µ 0 . Notice that λ is always less than or equal to 1. Thus, “small” values of λ are those less than some k < 1. Because n∑ n∑ (y i − µ 0 ) 2 = [(y i − y) + (y − µ 0 )] 2 i=1 = i=1 n∑ (y i − y) 2 + n(y − µ 0 ) 2 if k < 1, it follows that the rejection region, λ ≤ k, is equivalent to ∑ n i=1 (y i − y) 2 ∑ n i=1 (y i − µ 0 ) 2 < k2/n = k ′ i=1 ∑ n i=1 (y i − y) 2 ∑ n i=1 (y i − y) 2 + n(y − µ 0 ) 2 < k′ 1 1 + n(y − µ 0) 2 ∑ n i=1 (y i − y) 2 < k ′ . This inequality in turn is equivalent to n(y − µ 0 ) 2 ∑ n i=1 (y i − y) > 1 − 1 = k ′′ 2 k ′ n(y − µ 0 ) 2 >(n − 1)k 1 n∑ ′′ (y i − y) 2 n − 1 i=1 or, because y >µ 0 when λ √ (n − 1)k ′′ , n∑ (y i − y) 2 . i=1 Notice that √ n(Y − µ 0 )/S is the t statistic employed in previous sections. Consequently, the likelihood ratio test is equivalent to the t test of Section 10.8.

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