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

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ExampleSuppose X 1 ,...,X n are a sample from a population with mean µ,variance σ 2 , and third and fourth central moments µ 3 and µ 4 .Thenn 1/2 (s 2 −σ 2 ) ⇒ N(0,µ 4 −σ 4 )where ⇒ is notation for convergence in distribution.For simplicity I define s 2 = X 2 − ¯X 2 .Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 22 / 31
How to apply δ method1 Write statistic as a function of averages:◮ DefineW i =[ ]X2i.X i◮ See that [X2¯W n =]X◮ Definef(x 1 ,x 2 ) = x 1 −x 2 2◮ See that s 2 = f( ¯W n ).2 Compute mean of your averages:[ ] [µ W ≡ E( ¯W E(X2n ) = i) µ=2 +σ 2E(X i ) µ].3 In δ method theorem take Y n = ¯W n and y = E(Y n ).Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 23 / 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 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: The delta method pp 79-80, 131-135T
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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