Percentile bootstrap con…dence intervalsSuppose that a quantity = (F ) is of interest and thatTn= (the empirical distribution of Y1; Y2; : : : ; Yn)Based on B bootstrapped values T n1; T n2; : : : ; T nB, de…ne ordered valuesT n(1) T n(2) T n(B)Adopt the following convention (to locate lower and upper2points for thehistogram/empirical distribution of the B bootstrapped values). ForkL=j2(B + 1)kand kU= (B + 1) kL(bxc is the largest integer less than or equal to x) the intervalhT n(kL); T n(kU)i(1)contains (roughly) the “middle (1) fraction of the histogram of bootstrappedvalues.” This interval is called the (uncorrected) “(1 ) level bootstrap per-centile con…dence interval”for .The standard argument for why interval (1) might function as a con …denceinterval for is as follows. Suppose that there is an increasing function m()such that with = m() = m ( (F ))andb = m (Tn) = m ( (the emp irical distribution of Y1; Y2; : : : ; Yn))for large nb:s N; w2Then a con…dence interval for ishb zw;b + zwiand a corresponding con…dence interval for ishm1b zw; m1b + zwi(2)The argument is then that the bootstrap percentile interval (1) for large n andlarge B approximates this interval (2). The plausibility of an approximatecorrespondence between (1) and (2) might be argued as follows. Interval (2) ism1( zw) ; m1( + zw) hm1lower2point of the dsn ofb; m1upper2point of the dsn ofbi=hm1mlower2point of Tndsn; m1mlower2point of Tndsni=hlower2point of the dsn of Tn; upper2point of the dsn of Tni1and one may hope that interval (1) approximates this last interval. The beautyof the bootstrap argument in this context is that one doesn’t need to know thecorrect transformation m (or the standard deviation w) in order to apply
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