Environmental Engineering Reference
In-Depth Information
It is essential to verify whether the problem of “insufficient coverage” exists for the boot-
strap confidence intervals of δ
12
. The coverage probability refers to the probability for the
bootstrap confidence interval to cover the actual value of δ
12
. To investigate the sample size
under which the problem of insufficient coverage will be minimal, the probability for the
95% bootstrap confidence interval of δ
12
to cover the actual value of δ
12
is simulated. This
is done by the following steps:
1. Simulate (X
1
, X
2
) data of sample size =
n
from a bivariate standard normal distribution
with a chosen δ
12
.
2. Construct the 95% bootstrap confidence interval of δ
12
based on the simulated (X
1
,
X
2
) data.
3. See whether the 95% bootstrap confidence interval covers the actual of δ
12
.
Do this for 1000 independent realizations of (X
1
, X
2
) data. The coverage probability is
simply the ratio of successful coverage among the 1000 realizations. The bootstrap confi-
dence interval works properly if the coverage probability is close to 95%.
Table 1.7
shows the coverage probabilities for the 95% bootstrap confidence intervals of
δ
12
for three chosen values of δ
12
(0, 0.5, and 0.9). The numbers in the parentheses are for the
BCa method. It is clear that the coverage probability for δ
12
is close to 0.95 when
n
≥ 50. The
improvement brought about by the BCa method for the 95% bootstrap confidence intervals
of δ
12
is insignificant. It is therefore recommended that the percentile method is sufficient and
that the sample size
n
should be ≥50 for the bootstrap confidence intervals to work properly.
1.3.3.5 Goodness-of-fit test (the line test)
The normality of a column of data can be checked using the K-S test (see
Equations 1.23
through
1.26
). However, the bivariate normality of two columns of data requires a separate
check. The line test (Hald 1952; Kowalski 1970) is a reasonably simple test for bivariate
normality. The line test is based on the fact that if (X
1
, X
2
) are bivariate standard normal,
the variable Q
2
2
1
X
−
µ
X
−
µ
X
−
µ
X
−
µ
−
+
1
1
1
1
2
2
2
2
Q =
2
δ
(1.46)
12
1
−
δ
1
2
σ
σ
σ
σ
1
1
2
2
is χ-square distribution with 2 DOF. In reality, the value of μ, σ, and δ is not available.
Only the sample versions are available. Hence, (μ
1
, μ
2
) are replaced by (
m
1
,
m
2
), (σ
1
, σ
2
) are
replaced by (
s
1
,
s
2
), and δ
12
is replaced by its sample estimate (say using
Equation 1.40
). The
CDF for the 2-DOF χ-square distribution is
Table 1.7
Coverage probabilities for the 95% bootstrap confidence intervals of
δ
12
(Numbers in parentheses are for the BCa method)
δ
12
= 0
δ
12
= 0.5
δ
12
=
0.9
n
=
10
0.91 (0.94)
0.92 (0.94)
0.90 (0.93)
n
=
20
0.92 (0.93)
0.92 (0.93)
0.92 (0.92)
0.94 (0.95)
0.94 (0.94)
0.95 (0.95)
n
=
50
0.95 (0.95)
0.94 (0.94)
0.95 (0.95)
n
=
100
n
=
1000
0.95 (0.94)
0.94 (0.94)
0.94 (0.94)
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