I have a question regarding bootstrap coverage. I am trying to understand the
benefits of using the bootstrap for small sample sets. To do this I created a normal
population and then picked 10 from the populations and applied both traditional
statistical methods and the Bootstrap (bcanon, 5000 bootstrap samples) to calculate a
95% confidence interval of on the mean. I saved the width of the confidence interval
and how many misses, repeated this 1000 times, and output the summary on the CI width
and the number of misses for each method (actual script below) . I had expected to see
about 5% of the CI to miss the actual mean. For the traditional method about 6%
missed but for the Bootstrap it was over 11%.
Traditional:
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.4531 1.1460 1.3550 1.3690 1.5900 2.4330
[1] 0.062
Bootstrap:
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.3661 0.9455 1.1290 1.1380 1.3220 2.0160
[1] 0.113
The bootstrap method consistently missed almost twice as many times as the traditional
method.
What am I missing? Is this the best that can be expected when working with a sample
set of only 10 pieces?
Although this experiment was done using a normal distribution, my real datasets would
be non-normal of an unknown distribution.
Thanks for any help,
Art Nuzzo
Motorola
***************************************************************************************
library(bootstrap)
CImiss = function(M, lower, upper) {lower > M || upper < M }
CIr = function(lower, upper) {upper - lower}
C = c(); B = c() # CI Range
Ccov = 0; Bcov = 0 # Number of Ci Misses
cnt = 1000; # reps
x = rnorm(10000) # create population
m = mean(x)
for (i in 1:cnt) {
s = sample(x,10,replace=F) # sample population
tresults = t.test(s)
attach(tresults)
C[i] = CIr(conf.int[1],conf.int[2])
if (CImiss(m,conf.int[1],conf.int[2])) Ccov = Ccov + 1
detach(tresults)
bcaresults <- bcanon(s,5000,mean,alpha=c(.025,.975))
attach(bcaresults)
B[i] = CIr(confpoints[1,2],confpoints[2,2])
if (CImiss(m,confpoints[1,2],confpoints[2,2])) Bcov = Bcov + 1
detach(bcaresults)
}
print(summary (C))
print(Ccov/cnt)
print(summary (B))
print(Bcov/cnt)
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