AMDA-2008-Session 3

AMDA-2008-Session 3 - Empirical CDF and Statistical...

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Unformatted text preview: Empirical CDF and Statistical Functionals The Empirical Distribution Function ( 29 ( 29 ≤ ≤ = ≤ ≤ = ≤ = ∑ = x x x X I n x X I x i n i i i i 1 n i n X if X if 1 where n x points of no. ) ( F ˆ . X point data each at n 1 mass puts that CDF the is F ˆ function on distributi empirical The EDF… F(x) and n parameters with Binomial is ) ( F ˆ n of on distributi the x, fixed each for Then, F. cdf common a having variables random t independen are , , X Suppose n 1 x X n Properties of EDF . ) ) ( ) ( F ˆ P( , n As 4 1 ) ) ( ) ( F ˆ P( )) ( 1 )( ( ) ) ( ) ( F ˆ P( )) ( ˆ ( ) ) ( ) ( F ˆ P( 0, any for inequality s Chebyshev' By . 3 )) ( 1 )( ( )) ( F ˆ V( 2. ) ( )) ( F ˆ E( 1. n 2 n 2 n 2 n n n →- ∞ → ≤- ⇒- ≤- ⇒ ≤-- = = ε ε ε ε ε ε ε ε x F x n x F x n x F x F x F x x F V x F x n x F x F x x F x n Properties of EDF 2 2 x- x- n 2 ) ( ˆ ) ( sup P 0, any For : ) inequality (DKW Wolfowitz- Kiefer- Dvoretzky . 5 ) ( ) ( ˆ sup : Theorem Cantelli...
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This note was uploaded on 05/07/2009 for the course FIN AMDA taught by Professor Proflaha during the Spring '09 term at Indian Institute Of Management, Ahmedabad.

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AMDA-2008-Session 3 - Empirical CDF and Statistical...

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