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

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The definition p 75Def’n: A sequence of random variables X n converges in distributionto a random variable X ifE(g(X n )) → E(g(X))for every bounded continuous function g.TheoremThe following are equivalent:1 X n converges in distribution to X.2 P(X n ≤ x) → P(X ≤ x) for each x such that P(X = x) = 0.3 The limit of the characteristic functions of X n is the characteristicfunction of X: for every real tE(e itXn ) → E(e itX ).These are all implied by M Xn (t) → M X (t) < ∞ for all |t| ≤ ǫ for somepositive ǫ.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 6 / 31
Answering the questionsX n ∼ N(0,1/n) and X = 0. Then⎧⎨ 1 x > 0P(X n ≤ x) → 0 x < 0⎩1/2 x = 0Now the limit is the cdf of X = 0 except for x = 0 and the cdf of X isnot continuous at x = 0 so yes, X n converges to X in distribution.I asked if X n ∼ N(1/n,1/n) had a distribution close to that ofY n ∼ N(0,1/n).The definition I gave really requires me to answer by finding a limit Xand proving that both X n and Y n converge to X in distribution.Take X = 0. ThenandE(e tXn ) = e t/n+t2 /(2n) → 1 = E(e tX )E(e tYn ) = e t2 /(2n) → 1so that both X n and Y n have the same limit in distribution.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 7 / 31
Page 1 and 2: STAT 830Convergence in Distribution
Page 3 and 4: Convergence in Distribution p 72Und
Page 5: Some numerical examples?Answers dep
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 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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