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

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The medianSoP(ˆm ≤ x) = P( ∑ 1(X i ≤ x) ≥ k)i(= P √n(ˆFn (x)−F(x)) ≥ √ )n(k/n −F(x)) .From Central Limit theorem this is approximately( √n(k/n )−F(x))1−Φ √ .F(x)(1−F(x))Notice k/n → 1/2.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 30 / 31
MedianIf we put x = m+y/ √ n (where m is true median) we findF(x) → F(m) = 1/2.Also √ n(F(x)−1/2) → f(m) where f is density of F (if f exists).SoP( √ n(ˆm −m) ≤ y) → 1−Φ(−2f(m)y)That is,√ n(ˆm −1/2) → N(0,1/(4f 2 (m))).Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 31 / 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 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: Example - medianMany, many statisti

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

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