STAT 830 Convergence in Distribution - People.stat.sfu.ca - Simon ...

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Proof of CLTAs beforeE(e itn1/2¯X) → e −t2 /2 .This is the characteristic function of N(0,1) so we are done by ourtheorem.This is the worst sort of mathematics – much beloved of statisticians– reduce proof of one theorem to proof of much harder theorem.Then let someone else prove that.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 14 / 31
Edgeworth expansionsIn fact if γ = E(X 3 ) thenφ(t) ≈ 1−t 2 /2−iγt 3 /6+···keeping one more term.Thenlog(φ(t)) = log(1+u)whereu = −t 2 /2−iγt 3 /6+··· .Use log(1+u) = u −u 2 /2+··· to getlog(φ(t)) ≈ [−t 2 /2−iγt 3 /6+···]−[···] 2 /2+···which rearranged islog(φ(t)) ≈ −t 2 /2−iγt 3 /6+··· .Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 15 / 31
Page 1 and 2: STAT 830Convergence in Distribution
Page 3 and 4: Convergence in Distribution p 72Und
Page 5 and 6: Some numerical examples?Answers dep
Page 7 and 8: Answering the questionsX n ∼ N(0,
Page 9 and 10: Second graphN(1/n,1/n) vs N(0,1/n);
Page 11 and 12: Third graphN(1/sqrt(n),1/n) vs N(0,
Page 13: The Central Limit Theorem pp 77-79T
Page 17 and 18: RemarksThe error using the central
Page 19 and 20: Extensions of the CLT1 Y 1 ,Y 2 ,·
Page 21 and 22: The delta method pp 79-80, 131-135T
Page 23 and 24: How to apply δ method1 Write stati
Page 25 and 26: Delta Method ContinuesUsing binomia
Page 27 and 28: Alternative approachSuppose c is co
Page 29 and 30: Example - medianMany, many statisti
Page 31: MedianIf we put x = m+y/ √ n (whe

STAT 830 Convergence in Distribution - People.stat.sfu.ca - Simon ...

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