Application of the bootstrap to the analysis of vibration test data Page: 4 of 11
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Real World Bootstrap World
Unknown Probability Observed Observed Bootstrap
Distribution /Sample Distribution Samples
e=s(X) e* =s(X*)
Statistic /Bootstrap Replicates
of Interest f Statistic of Interest
Figure 3. The bootstrap approximation to the real world. The observed distribution is our best estimate of the true
distribution. The observed sample is X, and the statistic of interest 6 = s(X) can be computed based on this. In the
bootstrap world the observed data are used to generate as many bootstrap samples X* as we wish. Each bootstrap
sample is used in the formula 8* = s(X*) to compute a bootstrap replicate of the statistic of interest. The bootstrap
replicates are used to analyze the standard error, confidence intervals and bias of the statistical estimator.
A BOOTSTRAP EXAMPLE
Consider a set of data drawn from a random source with the probability density function illustrated in Figure 4.
One hundred data points are generated using a random source with this density.
We assume that the 100 points are characteristic of the source. The mean of the sampled points is 1.3440.
Using the bootstrap procedure outlined above, we create 400 bootstrap samples of these 100 points. (Normally, each
bootstrap sample contains as many points as are available in the original measured data set.) From each sample we
compute the sample mean. The standard deviation of these sample means is 0.0660; this is the standard error of the
mean estimate. The theoretical mean of this distribution is 1.3333. The 400 bootstrap replications of the original
data also allow computation of confidence intervals on our estimated mean. The 99% confidence intervals on the
mean are 1.1471 to 1.5073. The true mean lies well within these intervals. To further illustrate the typical
bootstrap results, Table I shows the results of seven different realizations of the distribution. In each case, the true
mean lies well within the confidence intervals indicated. Note that, as expected, a smaller number of points leads to a
broader confidence interval.
Table I.
x 0 x 1 Mean and confidence intervals for different realizations of
f(x) 3 the random data source shown in Figure 4.
2/3 --x+1 15 x <3 NPTS Mean Std. Err. Lower Upper True
Generated Estimate Mn. Est. 99%C.I. 99%C.I. Mean
100 1.3440 0.0660 1.1471 1.5073 1.3333
100 1.3571 0.0612 1.1325 1.5250 1.3333
50 1.3274 0.0881 1.1042 1.5677 1.3333
25 1.3250 0.0898 1.0513 1.5287 1.3333
0 NPTS Mean Std. Err. Lower Upper True
0 1 3 Generated Estimate Mn. Est. 90%C.I. 90%C.I. Mean
x 25 1.3761 0.1181 1.1862 1.5898 1.3333
50 1.2419 0.0903 1.1046 1.3949 1.3333
Figure 4. A random data source. 100 1.3607 0.0635 1.2605 1.4706 1.3333
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Hunter, N. F. & Paez, T. L. Application of the bootstrap to the analysis of vibration test data, article, August 1, 1995; Albuquerque, New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc623281/m1/4/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.