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

More documents

Recommendations

Info

Delta Method Continues7 Compute derivative (gradient) of f: has components (1,−2x 2 ).Evaluate at y = (µ 2 +σ 2 ,µ) to geta t = (1,−2µ).This leads ton 1/2 (s 2 −σ 2 ) ≈ n 1/2 [1,−2µ][X 2 −(µ 2 +σ 2 )¯X −µ]which converges in distribution to(1,−2µ)MVN(0,Σ).This rv is N(0,a t Σa) = N(0,µ 4 −σ 4 ).Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 26 / 31
Alternative approachSuppose c is constant. Define Xi ∗ = X i −c.Sample variance of Xi ∗ is same as sample variance of X i .All central moments of Xi ∗ same as for X i so no loss in µ = 0.In this case:[a t µ4 −σ= (1,0) Σ =4 ]µ 3µ 3 σ 2 .Notice thata t Σ = [µ 4 −σ 4 ,µ 3 ] a t Σa = µ 4 −σ 4 .Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 27 / 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: Delta Method ContinuesUsing binomia
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 ...

Create successful ePaper yourself

Delete template?

Save as template?