ABSTRACT
The basis of most large-sample approximations are based on the central limit theorem (Theorem 4.2.4), which states that if {Xn } be a sequence of independent, identically distributed random variables with mean θ and finite variance σ2, then Y n = ∑ i = 1 n ( X i − θ ) σ n = n ( X ¯ n − θ ) σ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2593.tif"/> where X ¯ n = ( 1 / n ) ∑ i = 1 n X i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2594.tif"/> , is approximately normal with mean 0 and unit variance as n ⟶ ∞. In other words, () L t n → ∞ F Y n ( y ) = ∫ − ∞ y 1 2 π e − 1 / 2 x 2 d x = ϕ ( y ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2595.tif"/> or we say that X ¯ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2596.tif"/> is asymptotically normal with mean θ and variance σ2/n. This approximation depends on the first two moments of X ¯ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2597.tif"/> . Theorem
371Let g ( X ¯ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2598.tif"/> be any function of X ¯ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2599.tif"/> with nonzero derivative g’(θ) at X ¯ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2600.tif"/> = θ. Then n ( g ( X ¯ n ) − g ( θ ) ) σ g ′ ( θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2601.tif"/> is asymptotically normal with zero mean and unit variance.
Proof. Expand g ( x ¯ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2602.tif"/> by Tayler series as g ( x ¯ n ) = g ( θ ) + ( x ¯ n − θ ) ( g ′ ( θ ) + ∊ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2603.tif"/> with ∊ n ⟶ 0 as g ( x ¯ n ) → g ( θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2604.tif"/> when n ⟶ ∞. This implies that for every small k there exists a δ such that | ∈ n | < k whenever | x ¯ n − θ | < δ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2605.tif"/>
Hence P { | ∈ n | < k } ≥ P { | X ¯ n − θ | < δ } → 1 as n → ∞ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2606.tif"/>
Since k: is arbitrary, {∊ n } converges in probability to zero. Now since n ( X ¯ n − θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2607.tif"/> is asymptotically normal with mean zero and variance σ2, and ∊ n ⟶ 0 in probability, n ( g ( X ¯ n ) − g ( θ ) ) − n ( X ¯ n − θ ) g ′ ( θ ) = n ( X ¯ n − θ ) ∈ n → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2608.tif"/> in probability. Hence n ( g ( X ¯ n ) − g ( θ ) ) and n ( X ¯ n − θ ) g ′ ( θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2609.tif"/> have the same asymptotic distribution which is normal with mean 0 and variance σ 2 ( g ′ ( θ ) ) 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2610.tif"/> . Q.E.D.
ExampleLet W be distributed as the central chi-square with n degrees of freedom. Then W = ∑ i = 1 n X i 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2611.tif"/> where X 1,…,Xn are independently, identically distributed normal random variables with mean 0 and variance 1. From (6.8), E ( X i 2 ) = 1 , var ( X i 2 ) = 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2612.tif"/> 372which implies that E(W) = n, var(W) = 2n. Hence the asymptotic distribution of W − n 2 n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2613.tif"/> is N(0,1).
ExampleLet X 1,…,Xn be independent, identically distributed Bernoulli random variables with parameter θ and let X ¯ n = 1 n ∑ i = 1 n X i . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2614.tif"/>
Then E ( X ¯ n ) = θ , var ( X ¯ n ) = θ ( 1 − θ ) n . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2615.tif"/>
Define g ( X ¯ n ) = X ¯ n ( 1 − X ¯ n ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2616.tif"/>
With g ( θ ) = θ ( 1 − θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2617.tif"/> , n ( g ( X ¯ n ) − g ( θ ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2618.tif"/> is asymptotically normally distributed with mean zero and variance θ(1 − θ)[g’(θ)]2 as n ⟶ ∞.
Since ( g ′ ( θ ) ) 2 = ( 1 − 2 θ ) 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2619.tif"/> , θ ( 1 − θ ) [ g ′ ( θ ) ] 2 = θ ( 1 − θ ) ( 1 − 2 θ ) 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2620.tif"/>
ExampleLet X 1,…,Xn be independent, identically distributed normal random variables with mean 0 and variance 1, and let F = ( 1 / k ) ∑ i = 1 k X i 2 [ 1 / ( n − k ) ] ∑ j = k + 1 n X j 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2621.tif"/>
From Section 6.1.17 Fis distributed as central F with k, n − k degrees of freedom. Let us assume k is fixed and n ⟶ ∞. This implies that n − k ⟶ ∞. By Theorem 4.1.3 (the weak law of large numbers) 1 ( n − k ) ∑ j = k + 1 n X j 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2622.tif"/> tends to 1 in probability as n ⟶ ∞.
By Slutsky’s theorem (Exercise 4 in Chapter 4), F is asymptotically distributed as (1/k) ( 1 / k ) χ k 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749920/87c2a07c-bbff-4506-b6ab-506959b79c01/content/eq2623.tif"/> as n ⟶ ∞.
