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

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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 .

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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 , τ ) for τ Θ, where Θ is a subset of some metric space. Remark 1 . In time series applications, generally, Θ = [0 , 1] Let 1 T ξ T ( τ ) = ( g t ( x t , τ ) Eg t ( x t , τ )) T t =1 ξ T ( τ ) is a random function; it maps each τ Θ to an n valued random variable. ξ T ( τ ) is called an empirical process. Under very general conditions, standard arguments show that ξ T ( τ ) converges pointwise, i.e. τ 0 Θ, ξ T ( τ 0 ) N (0 , σ 2 ( τ 0 )). Also, standard arguments imply that on a finite collection of points, ( τ 1 , ..., τ p ), ξ T ( τ 1 ) . . . N (0 , Σ( τ 1 , ..., τ p )) (1) ξ T ( τ p ) We would like to generalize this sort of result so that we talk about the convergence of ξ 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 ˆ T ( x ) = 1 1 ( x t x ) T Two possible statistics for testing whether F ˆ n ( x ) equals F ( x ) are the Kolmogorov-Smrinov statistic, sup n ( F ˆ n ( x ) F ( x )) x and the Cramer-von Mises statistic n ( F ˆ n ( x ) F ( x )) 2 dF ( x ) This fits into the setup above with 1 ξ T ( τ ) = T 1 ( x t τ ) F ( τ ) . For independent x t , finite dimensional covergence is easy to verify and for any τ 1 , τ 2 we have �� ξ T ( τ 1 ) F ( τ 1 )(1 F ( τ 1 )) F ( τ 1 ) F ( τ 2 ) F ( τ 1 ) F ( τ 2 ) ξ T ( τ 2 ) N 0 , F ( τ 1 ) F ( τ 2 ) F ( τ 1 ) F ( τ 2 ) F ( τ 2 )(1 F ( τ 2 )) Definition 3. We define a metric for functions on Θ as d ( b 1 , b 2 ) = sup τ Θ | b 1 ( τ ) b 2 ( τ ) |
Suﬃcient Conditions for Stochastic Equicontinuity 2 Definition 4. B = bounded functions on Θ Definition 5. U ( B ) = class of uniformly continuous (wrt d ()) bounded functionals from B to Example 6 . Examples of elements of U ( B ) include: Evaluation at a point: f τ 0 ( ξ ) = ξ ( τ 0 ) Integration: f ( ξ ) = Θ ξ ( τ ) Definition 7. convergence in B : ξ T ξ iff f ∈ U ( B ) we have Ef ( ξ T ) Ef ( ξ ) Remark 8 . This definition of convergence implies pointwise convergence. If ξ T ξ , then by definition for each τ 0 and k , T ( τ 0 ) k ( τ 0 ) k . Then, if the distribution of ξ ( τ 0 ) is completely determined by its moments (as it

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

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