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AMDA-2008-Session 4

# AMDA-2008-Session 4 - The Bootstrap Introduction The...

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The Bootstrap

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Introduction The bootstrap is a computational method for obtaining estimate of standard error of an estimator. It can also be used for obtaining confidence intervals The Bootstrap technique was invented by B. Efron in 1979.
Bootstrap Idea . simulation using ) ( V e Approximat : 2 Step ) ( V with ) ( V Estimate : 1 Step : steps two has idea bootstrap The . T of variance the be ) (T V and statistic a be ) ,..., ( T Let n n F ˆ F ˆ F n n F 1 n n n n n T T T X X g =

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Note n X X n T n T X T n i i n n n n = - = = = 1 2 F ˆ 2 F n F ˆ ) ( 1 ) ( V hence, ; ) ( V then T if example For form. simple a in expressed be can ) ( V if needed not is 2 Step n n σ
Law of Large Numbers )) ( ( ) ( n 1 Then, . h function any for ),... h(X ), h(X is so then variables random iid of sequence a be ,... , X if that Note : Corollary X Then . ) (X E with variables random iid of sequence a be ,... , X Let 1 2 1 2 1 n i F 2 1 X h E X h X X n i P i P = → → = μ μ

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Applications ) ( ) ( 1 ) ( 1 ) ( 1 ) ( 1 1 2 1 2 1 2 2 1 2 t F t X I n X V X X n X X n X E X n P n i i F P n i i n i i P n i i →
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AMDA-2008-Session 4 - The Bootstrap Introduction The...

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