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

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Slutsky’s Theorem p 75TheoremIf X n converges in distribution to X and Y n converges in distribution (or inprobability) to c, a constant, then X n +Y n converges in distribution toX +c. More generally, if f(x,y) is continuous then f(X n ,Y n ) ⇒ f(X,c).Warning: the hypothesis that the limit of Y n be constant is essential.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 20 / 31
The delta method pp 79-80, 131–135TheoremSuppose:Sequence Y n of rvs converges to some y, a constant.X n = a n (Y n −y) then X n converges in distribution to some randomvariable X.f is differentiable ftn on range of Y n .Then a n (f(Y n )−f(y)) converges in distribution to f ′ (y)X.If X n ∈ R p and f : R p ↦→ R q then f ′ is q ×p matrix of first derivatives ofcomponents of f.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 21 / 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: Extensions of the CLT1 Y 1 ,Y 2 ,·
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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