4.9. Bootstrap Procedures 309- Whilej≤B, do steps 3–6.
- Obtain a random sample with replacement of sizen 1 fromz.Callthesample
x∗′=(x∗ 1 ,x∗ 2 ,...,x∗n 1 ). Computex∗j. - Obtain a random sample with replacement of sizen 2 fromz.Callthesample
y∗′=(y 1 ∗,y∗ 2 ,...,y∗n 2 ). Computey∗j. - Computevj∗=y∗j−x∗j.
- The bootstrap estimatedp-value is given by
̂p∗=#Bj=1{v∗j≥v}
B. (4.9.10)
Note that the theory cited above for the bootstrap confidence intervals covers this
testing situation also. Hence, this bootstrapp-value is valid.
Example 4.9.2.For illustration, we generated data sets from a contaminated nor-
mal distribution, using the R functionrcn.LetW be a random variable with
the contaminated normal distribution (3.4.17) with proportion of contamination
=0.20 andσc= 4. Thirty independent observationsW 1 ,W 2 ,...,W 30 were gen-
erated from this distribution. Then we letXi=10Wi+ 100 for 1≤i≤15 and
Yi=10Wi+15+ 120 for 1≤i≤15. Hence the true shift parameter is Δ = 20. The
actual (rounded) data are
Xvariates
94.2 111.3 90.0 99.7 116.8 92.2 166.0 95.7
109.3 106.0 111.7 111.9 111.6 146.4 103.9
Yvariates
125.5 107.1 67.9 98.2 128.6 123.5 116.5 143.2
120.3 118.6 105.0 111.8 129.3 130.8 139.8Based on the comparison boxplots below, the scales of the two data sets appear to
be the same, while they-variates (Sample 2) appear to be shifted to the right of
x-variates (Sample 1).--------
Sample 1 ----I +I-- * O
------------------
Sample 2 * ------I + I--------
----------
+---------+---------+---------+---------+---------+------C3
60 80 100 120 140 160