BootstrapPercentile

# BootstrapPercentile - ² b ³ ² zw ³;m ± 1 ² b ³ zw...

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Suppose that a quantity = ± ( F ) is of interest and that T n = ± ( the empirical distribution of Y 1 ; Y 2 ;:::; Y n ) Based on B bootstrapped values T n 1 ;T n 2 ;:::;T nB T n (1) T n (2) T n ( B ) Adopt the following convention (to locate lower and upper 2 points for the histogram/empirical distribution of the B bootstrapped values). For k L = j ² 2 ( B + 1) k and k U = ( B + 1) ² k L ( b x c is the largest integer less than or equal to x ) the interval h T n ( k L ) ;T n ( k U ) i (1) contains (roughly) the ±middle (1 ² ² ) fraction of the histogram of bootstrapped values.² This interval is called the (uncorrected) ± (1 ² ² ) level bootstrap per- . interval for is as follows. Suppose that there is an increasing function m ( ± ) such that with ³ = m ( ) = m ( ± ( F )) and b ³ = m ( T n ) = m ( ± ( the empirical distribution of Y 1 ; Y 2 ;:::; Y n )) for large n b ³ : s N ³;w 2 ± ³ is h b ³ ² zw; b ³ + zw i is h m ± 1

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Unformatted text preview: ² b ³ ² zw ³ ;m ± 1 ² b ³ + zw ³i (2) The argument is then that the bootstrap percentile interval (1) for large n and large B approximates this interval (2). The plausibility of an approximate correspondence between (1) and (2) might be argued as follows. Interval (2) is ´ m ± 1 ( ³ ² zw ) ;m ± 1 ( ³ + zw ) µ ³ h m ± 1 ² lower ² 2 point of the dsn of b ³ ³ ;m ± 1 ² upper ² 2 point of the dsn of b ³ ³i = h m ± 1 ² m ² lower ² 2 point of T n dsn ³³ ;m ± 1 ² m ² lower ² 2 point of T n dsn ³³i = h lower ² 2 point of the dsn of T n ; upper ² 2 point of the dsn of T n i 1 and one may hope that interval (1) approximates this last interval. The beauty of the bootstrap argument in this context is that one doesn&t need to know the correct transformation m (or the standard deviation w ) in order to apply it. 2...
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BootstrapPercentile - ² b ³ ² zw ³;m ± 1 ² b ³ zw...

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