Percentile bootstrap con…dence intervals
Suppose that a quantity
= (F ) is of interest and that
Tn = (the empirical distribution of Y1 ; Y2 ; : : : ; Yn )
Based on B bootstrapped values Tn1 ; Tn2 ; : : : ; TnB , de…ne ordered values
Tn(B)
Tn(2)
Tn(1)
Adopt the following convention (to locate lower and upper 2 points for the
histogram/empirical distribution of the B bootstrapped values). For
j
k
kL =
(B + 1) and kU = (B + 1) kL
2
(bxc is the largest integer less than or equal to x) the interval
h
i
Tn(kL ) ; Tn(kU )
(1)
contains (roughly) the “middle (1
) fraction of the histogram of bootstrapped
values.” This interval is called the (uncorrected) “(1
) level bootstrap percentile con…dence interval” for .
The standard argument for why interval (1) might function as a con…dence
interval for is as follows. Suppose that there is an increasing function m( )
such that with
= m( ) = m ( (F ))
and
b = m (Tn ) = m ( (the empirical distribution of Y1 ; Y2 ; : : : ; Yn ))
for large n
Then a con…dence interval for
h
:
bs
N
; w2
is
b
zw; b + zw
i
and a corresponding con…dence interval for is
h
i
m 1 b zw ; m 1 b + zw
(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(
zw) ; m 1 ( + zw)
i
h
point of the dsn of b ; m 1 upper
point of the dsn of b
m 1 lower
2
2
h
i
1
1
= m
m lower
point of Tn dsn ; m
m lower
point of Tn dsn
2
2
h
i
= lower
point of the dsn of Tn ; upper
point of the dsn of Tn
2
2
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