
60 Statistical consequences of what we learned so far
= P
max
b
F
n
(x)−
λ
√
n
,0
≤F(x)≤min
b
F
n
(x)+
λ
√
n
,1
for all x
.
The last equality is true because 0 ≤ F(x) ≤ 1. Therefore the random
corridor or “strip” with the boundaries for each x
max
b
F
n
(x) −
λ
√
n
,0
,min
b
F
n
(x) +
λ
√
n
,1
(6.1)
covers the distribution function Fwith probability K
n
(λ). This is the
desired confidence strip for F. If one chooses λ so that K
n
(λ) = 1 −α
with the given α or K(λ) = 1 −α, then F will be covered by this
confidence strip with probability 1 −α, or probability asymptotically
equal to 1 −α.
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 6.3 Distribution function F (smooth line) and