# Lecture 4 - Large Sample Elementary Multivariate...

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Elementary Multivariate Statistical Inference Large Sample II.A. Large Sample Properties of IID Data ( not necessarily normal ) 1. Note : “Large” means n - p is large. 2. Suppose X 1 , . . . , X n are i.i.d. data with mean μ and covariance matrix Σ . Then (a) Law of Large Number : X P -→ μ as n → ∞ (i.e., X is a consistent estimator of μ ). (b) S P -→ Σ as n → ∞ (i.e., S is a consistent estimator of Σ ). (c) Central Limit Theorem : X N μ , Σ n ! for large n. i.e., n ( X - μ ) N ( 0 , Σ ) . Hence d 2 ( X , μ ) χ 2 p , where d 2 ( X , μ ) = ( X - μ ) 0 Σ n ! - 1 ( X - μ ) = n ( X - μ ) 0 Σ - 1 ( X - μ ) . 3. For practical purposes, the following approximations are made when n - p is very large: (a) X N ( μ , S /n ) (estimate Σ by S ), (b) n ( X - μ ) 0 S - 1 ( X - μ ) = d 2 ( X , μ ) χ 2 p . 4. Example : Find a 95% confidence region for μ = " μ 1 μ 2 # if n = 78 observations give summary data

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Unformatted text preview: X = " 2 1 # and S = " 10-2-2 10 # . Solution : We know P ( d 2 ( X , μ ) < c 2 ) = . 95 if c 2 = χ 2 2 ( . 05) = 5 . 99 , ( X-μ ) S n !-1 ( X-μ ) ≤ c 2 . eigenvectors of S n eigenvalues of S n : λ i n ( same as those of S ) ( eigenvalues of S : λ i ) u 1 = " . 707-. 707 # ( λ 1 = 12) λ 1 n = 12 78 u 2 = " . 707 . 707 # ( λ 2 = 8) λ 2 n = 8 78 1 2 3 4-1 1 2 3 μ 1 μ u X + half axis length = c s λ i n We are 95% conﬁdent that μ is in the ellip-soid centered at X = " 2 1 # with axes . 960 u 1 & . 784 u 2 . II.A-2...
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