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

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Purposes of These NotesDefine convergence in distributionState central limit theoremDiscuss Edgeworth expansionsDiscuss extensions of the central limit theoremDiscuss Slutsky’s theorem and the δ method.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 2 / 31
Convergence in Distribution p 72Undergraduate version of central limit theorem:TheoremIf X 1 ,...,X n are iid from a population with mean µ and standarddeviation σ then n 1/2 (¯X −µ)/σ has approximately a normal distribution.Also Binomial(n,p) random variable has approximately aN(np,np(1−p)) distribution.Precise meaning of statements like “X and Y have approximately thesame distribution”?Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 3 / 31
Page 1: STAT 830Convergence in Distribution
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 and 14: The Central Limit Theorem pp 77-79T
Page 15 and 16: Edgeworth expansionsIn fact if γ =
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

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