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Unformatted text preview: The CLT does not state that means of samples have normal distributions. ˜˜˜™™ ™ 19/42 Part 11: Asymptotic Distribution A Central Limit Theorem ˜˜˜™™ ™ 20/42 μ σ μ → σ 1 n 2 d LindebergLevy CLT (the simplest version of the CLT) If x ,..., x are a random sample from a population with finite mean and finite variance , then n(x ) N(0,1) Note, not the limitin = σ μ → n d n g distribution of the mean, since the mean, itself, converges to a constant. A useful corollary: if plim s , and the other conditions are met, then n(x ) N(0,1) s Part 11: Asymptotic Distribution LindebergLevy vs. LindebergFeller LindebergLevy assumes random sampling – observations have the same mean and same variance. LindebergFeller allows variances to differ across observations, with some necessary assumptions about how they vary. Most econometric estimators require Lindeberg Feller (and extensions such as Lyapunov). ˜˜˜™™ ™ 21/42 Part 11: Asymptotic Distribution Order of a Sequence Order of a sequence ‘Little oh’ o(.). Sequence hn is o(n) (order less than n) iff n hn 0. Example: hn = n1.4 is o(n1.5) since n1.5 hn = 1 /n.1 0. ‘Big oh’ O(.). Sequence hn is O(n) iff n hn a finite nonzero constant. Example 1: hn = (n2 + 2n + 1) is O(n2). Example 2: ixi2 is usually O(n1) since this is nthe mean of xi2 and the mean of xi2 generally converges to E[xi2], a finite constant. What if the sequence is a random variable? The order is in terms of the variance. Example: What is the order of the sequence in random sampling? Var[ ] = σ2/n which is O(1/n) ˜˜˜™™ ™ 22/42 n x n x Part 11: Asymptotic Distribution Asymptotic Distribution An asymptotic distribution is a finite sample approximation to the true distribution of a random variable that is good for large samples, but not necessarily for small samples. Stabilizing transformation to obtain a limiting distribution. Multiply random variable xn by some power, a, of n such that the limiting distribution of naxn has a finite, nonzero variance. Example, has a limiting variance of zero, since the variance is σ2/n. But, the variance of √n is σ2. However, this does not stabilize the distribution because E[ ] = √ nμ. The stabilizing transformation would be ˜˜˜™™ ™ 23/42 n x n x n n x n(x ) μ Part 11: Asymptotic Distribution Asymptotic Distribution Obtaining an asymptotic distribution from a limiting distribution Obtain the limiting distribution via a stabilizing transformation Assume the limiting distribution applies reasonably well in finite samples Invert the stabilizing transformation to obtain the asymptotic distribution Asymptotic normality of a distribution....
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 Fall '10
 H.Bierens
 Econometrics, Central Limit Theorem, Normal Distribution, Variance, Probability theory, asymptotic distribution, Slutsky Theorem

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