Inferences for a Mean Vector
Inferences for a Mean Vector concerning hypothesis testing, random sampling, confidence intervals, Hotelling's T2 test, likelihood ratio tests, and more are explored in this comprehensive content. The content dives into statistical inferences and procedures related to mean vectors through various analytical methods and examinations.
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Inference Concerning a Mean ( ) 1 n 1 n n ( ) 2 = = 2 2 2 ,..., ~ , , independent X X NID X X S X X X S 1 n i i 1 n = = 1 1 i i ( ) 2 1 n S 2 X Z ( ) = 2 n 2 df ~ , ~ 0,1 ~ ~ if X N Z N t Z 1 df 2 n n 2 df df X = n X S X S = = ~ 1 T t P t t 1 1 1 n n n ( ) 2 2 n n 2 1 n S ( ) 1 n 2 ( ) 1 n 1 n n 2 = = 2 Random Sample: ,..., x x x x s x x 1 n i i 1 n = = 1 x s 1 i i ( ) = = = : : : : 2 0 H H TS t RR t t P P t t 0 0 0 1 1 A obs obs n n obs 2 n ( )( ) ( s ) 1 ( ) = = 2 obs 2 2 n Squaring : Reject if t H t n x x t F 0 0 0 1 1, 1 obs n 2 s ( ) x t 1 100% Confidence Interval for : 1 n 2 n
Inference for a Mean Vector Hotellings T2- I M L X 1 1 11 M O 1 M j p ( ) = = = X X ,..., ~ , M NID X 1 n j L X 1 jp p p pp ( )( ) 1 n 1 n n ( ) ( ) = = X X S X X X X S ' 1 ~ Wishart , independent X S n W , 1 , 1 j j j p n p n 1 n = = 1 1 j j ( ) ( ) = Under : : ~ , H n N X 0 0 0 0 p 1 W df n p p ( ) ( ) ( ) , p df df = 2 0 ' 0 , , ~ when , T N N p n p F N 0 W , , p p p p df ( ) 1 ( ) p 1 S 1 1 n n n p ( ) ( ) ( ) ( ) = = 1 X ' 1 ~ df n n n n p n p F X X 'S X , n n n p 2 ~ T p n p F ( ) , 1 p ( ) p 1 n n p ( ) ( ) ( X S ( ) ) n n p ( ) ( ) = = = 2 1 1 X S X ' ' P T n p n p F P n , p n p F X ( ) , 1 p
Inference for a Mean Vector Hotellings T2- II ( )( ) 1 n 1 n n = = x x x x S x x x x Random Sample: ,..., ' 1 n j j j 1 n = = 1 1 j j ( ) ( ) = = 2 1 : : : ' H H TS T n x S x 0 0 0 0 0 A ( ) p 1 n n p n n p ( ) ( ) 2 2 : equivalently: RR T p n p F T p n p F ( ) , , 1 p n n p = 2 P F s invariant to (full rank) linear transformations of P T ( ) , p n p 1 p ( ) C = + = 2 X Y C X d i : rank T p p p 1 1 1 p p p 1 p Y Y = = + = = E V C d C C' Y X Y Y = + = + C d y Cx d Y 0 0 ( ) ( ) ( ) )( ( ( )( ) 1 1 n n = = = S C x x C x x ' C x x x x 'C' CSC' y j j j j 1 1 n n = = 1 1 ( ( j j ( ) ( ) ( ) ( ) ) ) ( ) 1 = + + + + = 2 Cx d ) ' C' C' C T n d ' CSC' Cx d C ( x d 0 0 y ( ) ) ) ( 1 = = 1 1 1 2 x ' n n T S C C x x S 0 0 0 0
T2and Likelihood Ratio Tests ( ) = X X ,..., ~ , : : Random Sample: ,..., NID H H x x 1 0 0 0 1 n A n ( )( ) 1 n n ^ = Maximum Likelihood (ML) Estimate of : ' x x x x j j = 1 j 1 n n ( )( ) ^ = Under null hypothesis: ' x x 0 0 0 j j = 1 j /2 /2 n n ^ ^ ( ) max , L ( , 0 0 = = = Likelihood Ratio: ) ^ ^ max L 0 , n ( )( ) ( ) ^ x 1 ' n x 1 0 0 j j 2 T n ( ) = 1 j = = + = 2/ 2 n Wilks' Lambda: 1 1 T n ^ 1 ( )( ) n x x x x ' 0 j j = 1 j
Confidence Regions for Mean Vector - I ( n ) 1 p n ( ) ( ) ( ) ( ) n p ( ) = 1 1 X 'S ~ 1 n p n p F P n p n p F X X 'S X ( ) , , 1 p p n ( )( ) 1 n 1 n n = = x x x x S x x x x ' Sample Measurements: ,..., 1 n i i i 1 n = = 1 1 i i ( ) ( ) n p ( ) ( ) 1 x 'S 1 100% Confidence Interval for : All such that n p n p F x ( ) , 1 p n ( ) ) 1 p p n n n ( ) ( ) ( ) 1 Equivalently: All such that p n p F x 'S x ( , ( ) ) 1 p p n n n ( ) x e Confidence Ellipsoid centered at with axes: p n p F ( , i i e ues and eigenvectors of S where and are the eigenval i i
Confidence Regions for Mean Vector - II ( ) ( ) n p 1 X 'S ~ n p n p F X ( ) , 1 p n ( ) ( ) n p ( ) = 1 X 'S 1 P n p n p F X ( ) , 1 p n a In general fixed : ( ) ) ( ) ) 1 p 1 p p n n n p n n n ( ) ( ) + = a'X a'Sa a' 1 P p n p F p n p F a'X a'Sa ( ( , , ( ) 2 x'd x'Bx = = 1 1 d'B d x B d "Proof:" Maximization Lemma : max x 0 with max @ 0 c c = = = x a d X B S ( ) ( ) ( a'Sa ) ( ) 2 2 a' X n a' X ( ) ( ) = = = 1 2 X 'S max a max a n n T X a'Sa ( ) = 1 a S X with max @
Confidence Regions for Mean Vector - III ( ) ) ( ) ) 1 p 1 p p n n n p n n n ( ) ( ) + = a'X a'Sa a' 1 P p n p F p n p F a'X a'Sa ( ( , , ( )( ) 1 n 1 n n = = x x x x S x x x x ' Sample Measurements: ,..., 1 n j j j 1 n = = 1 1 j j = = a ' a ' Defining: 1 0 0 ... 0 0 1 1 p ( n ) 1 p n s n ( ) with confidence coefficient 1 will contain ii x p n p F ( ) i , i p = 1 0 a' For differences among means: : e.g. with 1 0 : 1 2 i k ( n ) + 1 p n ( ) 2 s s n s ( ) will contain with confidence coefficient 1 ii ik kk x x p n p F ( ) i k , i k p ( ) ( ) ( ) 1 100% Confidence Ellipsoid for , : Set of , such that: i k i k ( ( ) ) ( n ) x 1 s s s s 1 p n ( ) ( ) i i ( ) ii ik n x x p n p F ( ) i k , i k p x ik kk k k
Simultaneous CIs and Large-Sample Inference = Bonferroni Inequality: Simultaneous Inferences (Tests/CI's): m ,..., with ( False) C C P C 1 m i i m m ( ) ( ) = = = + + (All True) 1 (At least one False) 1 ( False) 1 1 ( True) 1 ... P C P C P C P C 1 i i i i m = = 1 1 i i = = Setting (All True) 1 1 P C m i i m m ( ) 1 100% Bonferroni Simultaneous Confidence Intervals for : i s n = 1,..., ii x t i p i 1 n 2 p X n X = = X X Large-Sample Inferences: Sa mple measurements: ,..., ,..., with large n with (positive definite) E V 1 n i X i X x x : p 1 ( ) ( ) ( ) = 1 2 p Reject : if H n x 'S x 0 0 0 0 a'Sa a'Sa ( ) ( ) + 2 p 2 p a'X a' 1 P a'X a n n a'Sa ( ) ( ) 2 p a' Approximate 1 100% CI for : a'x n
