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

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SummaryTo derive approximate distributions:Show sequence of rvs X n converges to some X.The limit distribution (i.e. dstbn of X) should be non-trivial, like sayN(0,1).Don’t say: X n is approximately N(1/n,1/n).Do say: n 1/2 (X n −1/n) converges to N(0,1) in distribution.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 12 / 31
The Central Limit Theorem pp 77–79TheoremIf X 1 ,X 2 ,··· are iid with mean 0 and variance 1 then n 1/2¯X converges indistribution to N(0,1). That is,P(n 1/2¯X ≤ x) →1√2π∫ x−∞e −y2 /2 dy .Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 13 / 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: Third graphN(1/sqrt(n),1/n) vs N(0,
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

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

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