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sta 2006 0708_chapter_6

# sta 2006 0708_chapter_6 - STA 2006 Condence Intervals Large...

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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 } = 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

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Example 1: Suppose X 1 , . . . , X n are i.i.d. sampled from exponential distri- bution with mean 1 . Find the (approximate) 100(1 - α )% C.I. of λ when n is large. Solution: Note that E [ X 1 ] = 1 and V ar ( X 1 ) = 1 2 . By CLT, n ( ¯ X n - 1 ) 1 is approximately N (0 , 1) when n is large. That is, when n is large, Pr {- z α/ 2 n ( ¯ X n - 1 ) 1 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. of p when n is large. Solution: Note that E [ I 1 ] = p and V ar ( I 1 ) = p (1 - p ). By CLT, n ( ¯ I n - p ) p (1 - p ) is approximately N (0 , 1) when n is large. Note that ¯ I n is the same as ˆ p in the overview chapter and it is the maximum likelihood estimate of p . When n is large, the C.I. of p is derived by using Pr {- z α/ 2 n ( ¯ I n - p ) p (1 - p ) z α/ 2 } ≈ 1 - α Pr { n ( ¯ I n - p ) 2 z 2 α/ 2 p (1 - p ) } ≈ 1 - α.
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sta 2006 0708_chapter_6 - STA 2006 Condence Intervals Large...

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