This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: STA 2006 Confidence Intervals: Large Sample Approximation 20072008 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. (nonnormal) with common mean μ and variance σ 2 , (1) is still true by the Central Limit Theorem (CLT). Theorem 5.41 in HT:(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....
View
Full
Document
This note was uploaded on 06/05/2011 for the course STATISTICS 2006 taught by Professor Ho during the Spring '11 term at CUHK.
 Spring '11
 Ho
 Statistics

Click to edit the document details