CH5.InferenceMeanVar.pdf - Chapter 5 Inferences about a...

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Chapter 5. Inferences about a mean vector 1. Hotelling’s T 2 Univariate case Let X 1 , . . . , X n IR be a random sample with E( X 1 ) = µ. We want to test H 0 : µ = µ 0 and H 1 : µ ̸ = µ 0 . A standard statistic is the t statistic defined by t = ¯ X µ 0 s/ n where ¯ X = n i =1 X i /n and S 2 = n i =1 ( X i ¯ X ) 2 / ( n 1) . If X i follows the normal distribution, then t has a t - distribution with n 1 degree of freedom (d.f.) under H 0 : µ = µ 0 . In general, we reject H 0 when | t | is large (i.e. | t | > t n 1 ( α/ 2)). The equivalent testing procedure is to reject H 0 when t 2 is large (i.e. t 2 > t 2 n 1 ( α/ 2)). (Recall t 2 n 1 ( α/ 2) = F 1 ,n 1 ( α ).) Note that t 2 = n ( ¯ X µ 0 )( S 2 ) 1 ( ¯ X µ 0 ) . Also, 100 × (1 α )% CI is given as { µ : n ( ¯ X µ )( S 2 ) 1 ( ¯ X µ ) F 1 ,n 1 ( α ) } . 1
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Extension to multivariate data Let X 1 , . . . , X n IR p be a random sample with E( X 1 ) = µ . We want to test H 0 : µ = µ 0 and H 1 : µ ̸ = µ 0 . A natural generalization of t 2 is T 2 0 = n ( ¯ X µ 0 ) S 1 ( ¯ X µ 0 ) . This statistic is called the Hotelling’s T 2 . If X i follows the multivariate normal distribution, then, under H 0 : µ = µ 0 , T 2 0 ( n 1) p n p F p,n p where F p,n p is the F distribution with the d.f.s p and n p. Hence, we reject H 0 when T 2 0 > ( n 1) p n p F p,n p ( α ) . Note that when p = 1 , F 1 ,n 1 ( α ) = t 2 n 1 ( α/ 2) . 2
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Invariance to the affine transformation Let T 2 ( X ) be the Hotelling’s T 2 statistic based on the data X 1 , . . . , X n . Let Y i = CX i + d where C is p × p matrix and d IR p . Suppose that C is nonsingular. Then, T 2 ( Y ) is the same as T 2 ( X ) . For example, the Hotelling’s T 2 based on the Centigrade is the same as that based on the Fahrenheit. Hotelling’s T 2 spectral S eigenvalue decomposition S = p X i =1 l i e i e T i l 1 l 2 ≥ · · · ≥ l p . µ 0 = 0 T 2 = ( n ¯ X T ) S 1 ( n ¯ X ) = ( n ¯ X T ) ( p X i =1 l 1 i e i e T i ) ( n ¯ X ) = p X i =1 1 l i ( n ¯ X T e i )(
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