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Empirical Process Proof of the Asymptotic Distribution of Sample

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Empirical Process Proof of the Asymptotic Distribution of Sample Spring 1998 John Rust Economics 551b 37 Hillhouse, Rm. 27 Empirical Pro cess Pro of of the Asymptotic Distribution of Sample Quantiles th ~ De nition: Given 2 0; 1, the quantile of a random variable X with CDF F is de ned by: 1 F = inf fx j F x g: th th Note that is the median, is the 25 p ercentile, etc. Further if we de ne the 0 quantile :5 :25 as = lim and de ne similarly, it is easy to see that these are the lower and upp er 0 1 !0 ~ ~ p oints in the supp ort of X i.e. the minimum and maximum p ossible values of X which might ~ be 1 and +1 if X has unb ounded supp ort. Note also that if F is strictly increasing in a 1 neighb orho o d of , then = F is the usual inverse of the CDF F . If F happ ens to have \ at" sections, sayaninterval of p oints x satisfying F x= , then is the smallest x in this interval. The following lemma, a slightly mo di ed version of a lemma from R. J. Ser ing, 1980 Approximation Theorems of Mathematical Statistics Wiley, New York, provides some 1 basic prop erties of the quantile function F : 1 Lemma 1: Let F be a CDF. The quantile function F , 2 0; 1 is non-decreasing and left continuous, and satis es: 1 1. F F x x; 1 <x<1 1 2. F F ; 0 < <1 1 1 3. If F is strictly increasing in a neighb orho o d of = F wehave: F F = and 1 F F = . 1 4. F x if and only if x F . ~ ~ De nition: Let X ;:::;X bearandom sample of size N from a CDF F . Then the sample 1 N quantile ^ , 2 0; 1 is de ned by: 1 ^ = F ; N where F is the empirical CDF de ned by: N N X 1 ~ F x= I fX xg: N i N i=1 1 1 1 ~ ~ Thus ^ = F :5 is the sample median F :5 = medX ;:::;X , and ^ = F 0 is :5 1 N 0 N N N 1 1 ~ ~ the sample minimum, F 0 = minX ;:::;X , and ^ = F 1 is the sample maximum, 1 N 1 N N 1 ~ ~ F 1 = maxX ;:::;X . Since empirical CDF's have jumps of size 1=N unless more than 1 N N ~ one of the fX g's take the same value, then we can b ound the maximum di erence b etween i 1 and F F in Lemma 1-2 as follows: 1 ~ ~ Lemma 2: Let X ;:::;X bearandom sample from a CDF F and suppose that in this 1 N ~ ~ ~ ~ sample each X happens to be distinct, so that by reindexing we have X < X < < X < i 1 2 N1 ~ X . Then for al l 2 0; 1 we have: N 1 1 jF F j : N N N The following theorem shows that the asymptotic distribution of the sample quantiles ^ for 2 0; 1 are normally distributed. It is imp ortant to note that we exclude the two cases =0 and = 1 in this theorem since the asymptotic distribution of these extreme value statistics is very di erent and generally non-normal. ~ ~ Theorem: Let X ;:::;X be IID draws from a CDF F with continuous density f . Then if 1 N f > 0, we have: p p 1 1 2 N ^ = N F F = N0; ; N where: 1 2 = : 2 f Pro of: The Central Limit Theorem for IID random variables implies that for any x in the supp ort of F wehave: p 2 N F x F x = N 0; ; N 2 1 where = F x[1 F x]. Letting x = = F and using Lemma 1-3 wehave: p p 1 1 N F F = N F F F F = N 0; 1 : N N Furthermore, the prop ertyofstochastic equicontinuity from the theory of empirical processes see D. Andrews, 1996 Handbook of Econometrics vol. 4 for an accessible intro duction and de nition of sto chastic equicontinuity, wehave that the result given ab ove is una ected if we replace by a consistent estimate ^ : p p 1 1 N F ^ F ^ = N F F F F = N 0; 1 : N N N N Now note that Lemma 1-2 implies that p p 1 1 1 N F F N F F F F : N N N N ~ However since the true CDF F has a density, the probability of observing duplicate fX g's is i zero, so Lemma 2 implies that with probability1 wehave: p p p 1 1 1 N F F F F = N F F + O 1= N ; N p N N N which implies that: p 1 = N 0; 1 : N F F N 1 Nowwe apply the Delta theorem, i.e. wedoaTaylor series expansion of F F ab out the N 1 limiting p oint = F to get: h i 1 1 1 1 F F = F F + f ~ F F ; N N 2 where ~ is a p oint on the line segmentbetween ^ and . Using the result ab ove and Lemma 1-3 wehave: ! 1 p p F F 1 1 2 N N F F = N = N 0; ; N f ~ where wehave used Slutsky's Theorem and the fact that ~ ! since ^ ! with probability 1. 3.
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