Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 17

250 250

s = 23.82

m = 101.84

200 200

150 150

100 100

50 50

0 0

50 100

Bootstrap samples of m

150 0 10 20 30 40

Bootstrap samples of s

Figure 1.10 Bootstrap samples of m and s . The arrows indicate the point estimates.

(Y (1) , Y (2) , …, Y ( n ) ) represent the true population, and this assumption breaks down when n
is small, leading to the problem of insufficient coverage. Efron (1987) proposed an improved
bootstrap method consisting of bias corrections and acceleration adjustments (BCa) to miti-
gate this problem. An easy-to-follow guide for the BCa method can be found in Carpenter and
Bithell (2000). Using the BCa method, the 95% bootstrap confidence interval for σ becomes
[18.65, 29.91] for our case with n = 10. This confidence interval is closer to the analytical one
[16.38, 43.49] based on Equation 1.22 . However, the difference is still large.
To investigate the sample size under which the problem of insufficient coverage will be
minimal, the probability for the 95% bootstrap confidence interval to cover the actual value
of σ is simulated. This is done by the following steps:

1. Simulate (Y (1) , Y (2) , …, Y ( n ) ) from a normal distribution with μ = 100 and σ = 20.

2. Construct the 95% bootstrap confidence interval of μ (or σ) based on (Y (1) , Y (2) , …,
Y ( n ) ).

3. See whether the 95% bootstrap confidence interval covers the actual value of μ = 100
(or the actual value of σ = 20).

Do this for 1000 independent realizations of (Y (1) , Y (2) , …, Y ( n ) ). 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 1.5 shows the coverage probabilities for the 95% bootstrap confidence intervals of μ
and σ. The numbers in the parentheses are for the BCa method. It is clear that the coverage

Table 1.5 Coverage probabilities for the 95% bootstrap
confidence intervals of μ and σ (numbers in parentheses
are for the BCa method)
Sample size n μ σ

0.91 (0.91) 0.79 (0.85)

n = 10

n = 20 0.92 (0.92) 0.86 (0.90)

0.93 (0.93) 0.89 (0.92)

n = 50

0.93 (0.94) 0.93 (0.94)

n = 100

n = 1000 0.95 (0.95) 0.94 (0.95)