sta 2006 0708_chapter_6

sta 2006 0708_chapter_6 - STA 2006 Confidence Intervals:...

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Unformatted text preview: STA 2006 Confidence Intervals: Large Sample Approximation 2007-2008 Term II 1 Motivation In the previous chapters, X 1 , . . . , X n are i.i.d. sampled from N ( μ, σ 2 ). The C.I. of μ is derived by either considering Z = ¯ X- μ σ/ √ n ∼ N (0 , 1) (known σ case) (1) or T = ¯ X- μ S/ √ n ∼ t ( n- 1) (Unknown σ case) . (2) In this chapter, we relax the assumption of normal population. In general, the distribution of Z can be complex for small sample. However, if n is large and X 1 , . . . , X n are i.i.d. (non-normal) with common mean μ and variance σ 2 , (1) is still true by the Central Limit Theorem (CLT). Theorem 5.4-1 in H-T:(CLT) If X 1 , . . . , X n are i.i.d. with common mean μ and variance σ 2 , then Z n = √ n ( ¯ X n- μ ) σ converges in distribution to N (0 , 1) as n → ∞ . That is, for each x ∈ R , lim n →∞ Pr { Z n ≤ x } = Z x-∞ 1 √ 2 π exp {- z 2 2 } dz = Φ( x ) , (3) or equivalently, Z n d → Z where Z ∼ N (0 , 1). (or Z n converges to Z in distribution .) Remarks: • Application of CLT: if n is large, Pr { Z n ≤ z } ≈ Φ( z ). • The general definition of convergent in distribution: Let { X n } be a sequence of random variables and X be a random variable. Also, sup- pose F n and F are the c.d.fs of X n and X . Now for any x in the set of continuity points of F , if we have lim n →∞ Pr { X n ≤ x } = lim n →∞ F n ( x ) = F ( x ) = Pr { X ≤ x } , then X n is said to be converging in distribution to X . Such fact is denoted by X n d → X . To apply this fact, if n is large and x is in the set of continuity points of F , Pr { X n ≤ x } = F n ( x ) ≈ F ( x ). 1 Example 1: Suppose X 1 , . . . , X n are i.i.d. sampled from exponential distri- bution with mean 1 /λ . Find the (approximate) 100(1- α )% C.I. of λ when n is large. Solution: Note that E [ X 1 ] = 1 /λ and V ar ( X 1 ) = 1 /λ 2 . By CLT, √ n ( ¯ X n- 1 /λ ) 1 /λ is approximately N (0 , 1) when n is large. That is, when n is large, Pr {- z α/ 2 ≤ √ n ( ¯ X n- 1 /λ ) 1 /λ ≤ z α/ 2 } ≈ 1- α ⇔ Pr { - z α/ 2 √ n + 1 1 ¯ X n ≤ λ ≤ z α/ 2 √ n + 1 1 ¯ X n } ≈ 1- α Therefore, the approximate 100(1- α )% C.I. of λ is given by: [- z α/ 2 √ n + 1 1 ¯ x n , z α/ 2 √ n + 1 1 ¯ x n ]. The approximation can be evaluated by using the simulation program in ”Exp.R” for λ = 1 and target confidence level 95% with 100,000 replicates. n 5 10 20 50 100 500 Simulated C. L. 95 . 563% 95 . 482% 95 . 230% 95 . 176% 95 . 050% 95 . 017% Example 2: Suppose I 1 , . . . , I n are i.i.d. sampled from Bernoulli distribu- tion with probability of success p . Find the (approximate) 100(1- α )% C.I....
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This note was uploaded on 06/05/2011 for the course STATISTICS 2006 taught by Professor Ho during the Spring '11 term at CUHK.

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sta 2006 0708_chapter_6 - STA 2006 Confidence Intervals:...

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