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

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Delta Method Continues7 Take a n = n 1/2 .8 Use central limit theorem:n 1/2 (Y n −y) ⇒ MVN(0,Σ)where Σ = Var(W i ).9 To compute Σ take expected value of(W −µ W )(W −µ W ) tThere are 4 entries in this matrix. Top left entry is(X 2 −µ 2 −σ 2 ) 2This has expectation:E { (X 2 −µ 2 −σ 2 ) 2} = E(X 4 )−(µ 2 +σ 2 ) 2 .Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 24 / 31
Delta Method ContinuesUsing binomial expansion:E(X 4 ) = E{(X −µ+µ) 4 }= µ 4 +4µµ 3 +6µ 2 σ 2 +4µ 3 E(X −µ)+µ 4 .So Σ 11 = µ 4 −σ 4 +4µµ 3 +4µ 2 σ 2 .Top right entry is expectation ofwhich isSimilar to 4th moment get(X 2 −µ 2 −σ 2 )(X −µ)E(X 3 )−µE(X 2 )µ 3 +2µσ 2Lower right entry is σ 2 .So [µ4 −σΣ =4 +4µµ 3 +4µ 2 σ 2 µ 3 +2µσ 2 ]µ 3 +2µσ 2 σ 2Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 25 / 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 and 22: The delta method pp 79-80, 131-135T
Page 23: How to apply δ method1 Write stati
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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