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

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Towards precisionDesired meaning: X and Y have nearly the same cdf.But care needed.Q1: If n is a large number is the N(0,1/n) distribution close to thedistribution of X ≡ 0?Q2: Is N(0,1/n) close to the N(1/n,1/n) distribution?Q3: Is N(0,1/n) close to N(1/ √ n,1/n) distribution?Q4: If X n ≡ 2 −n is the distribution of X n close to that of X ≡ 0?Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 4 / 31
Some numerical examples?Answers depend on how close close needs to be so it’s a matter ofdefinition.In practice the usual sort of approximation we want to make is to saythat some random variable X, say, has nearly some continuousdistribution, like N(0,1).So: want to know probabilities like P(X > x) are nearlyP(N(0,1) > x).Real difficulty: case of discrete random variables or infinitedimensions: not done in this course.Mathematicians’ meaning of close: Either they can provide an upperbound on the distance between the two things or they are talkingabout taking a limit.In this course we take limits.Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 — Fall 2011 5 / 31
Page 1 and 2: STAT 830Convergence in Distribution
Page 3: Convergence in Distribution p 72Und
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 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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