rec07 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Empirical Process Theory 1 14.384 Time Series Analysis, Fall 2007 Recitation by Paul Schrimpf Supplementary to lectures given by Anna Mikusheva October 17, 2008 Recitation 7 Empirical Process Theory Let x t be a real-valued random k 1 vector. Consider some n valued function g t ( x t , ) for , where is a subset of some metric space. Remark 1 . In time series applications, generally, = [0 , 1] Let 1 T T ( ) = ( g t ( x t , ) Eg t ( x t , )) T t =1 T ( ) is a random function; it maps each to an n valued random variable. T ( ) is called an empirical process. Under very general conditions, standard arguments show that T ( ) converges pointwise, i.e. , T ( ) N (0 , 2 ( )). Also, standard arguments imply that on a finite collection of points, ( 1 ,..., p ), T ( 1 ) . . . N (0 , ( 1 ,..., p )) (1) T ( p ) We would like to generalize this sort of result so that we talk about the convergence of T (). Example 2 . Suppose you want to test whether x t has cdf F ( x ). The cdf of x t can be estimated by its empirical cdf, F T ( x ) = 1 1 ( x t x ) T Two possible statistics for testing whether F n ( x ) equals F ( x ) are the Kolmogorov-Smrinov statistic, sup n ( F n ( x ) F ( x )) x and the Cramer-von Mises statistic n ( F n ( x ) F ( x )) 2 dF ( x ) This fits into the setup above with 1 T ( ) = T 1 ( x t ) F ( ) . For independent x t , finite dimensional covergence is easy to verify and for any 1 , 2 we have T ( 1 ) F ( 1 )(1 F ( 1 )) F ( 1 ) F ( 2 ) F ( 1 ) F ( 2 ) T ( 2 ) N , F ( 1 ) F ( 2 ) F ( 1 ) F ( 2 ) F ( 2 )(1 F ( 2 )) Definition 3. We define a metric for functions on as d ( b 1 ,b 2 ) = sup | b 1 ( ) b 2 ( ) | Sucient Conditions for Stochastic Equicontinuity 2 Definition 4. B = bounded functions on Definition 5. U ( B ) = class of uniformly continuous (wrt d ()) bounded functionals from B...
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rec07 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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