The bootstrap principle is very simple.

**BOOTSTRAP PRINCIPLE**

In Chapter 2, we introduced the concept of
bootstrap sampling and told you that it was a nonparametric technique for
statistical inference. We also explained the mechanism for generating bootstrap
samples and showed how that mechanism is similar to the one used for simple
random sampling. In this section, we will de-scribe and use the bootstrap
principle to show a simple and straightforward method to generate confidence
intervals for population parameters based on the bootstrap samples. Reviewing
Chapter 2, the difference between bootstrap sampling and simple random
sampling is

1. Instead of sampling from a
population, a bootstrap sample is generated by sampling from a sample.

2. The sampling is done with
replacement instead of without replacement.

Bootstrap sampling behaves similarly to random
sampling in that each bootstrap sample is a sample of size *n* drawn at random from the empirical distribution *F _{n}*, a probability
distribution that gives equal weight to each observed data point (i.e., with
each draw, each observation has the same chance as any other observation of
being the one selected). Similarly, random sampling can be viewed as drawing a
sample of size

The bootstrap principle is very simple. We want to
draw an inference about the population mean through the sample mean. If we do
not make parametric assump-tions (such as assuming the observations have a
normal distribution) about the sam-pling distribution of the estimate, we
cannot specify the sampling distribution for inference (except approximately
through the central limit theorem when the esti-mate is a sample mean).

In constructing confidence intervals, we have
considered probability statements about quantities such as *Z* or *t* that have the form
( – *μ*)/*σ* or ( – *μ*)/*S*, where *σ* is the standard deviation or *S* is the estimated standard deviation for the sampling distribution (standard
error) of the estimated *X*. The bootstrap principle attempts to
mimic this process of constructing quantities such as *Z* and *t* and forming confidence
intervals. The sample estimate is replaced by its bootstrap analog *, the mean of a bootstrap sample. The parameter μ is replaced by .

Since the parameter *μ** *is unknown, we cannot actually calculate – *μ*, but from a bootstrap sample we can calculate * – *X*. We then
approximate the dis-tribution of * – by generating many bootstrap
samples and computing many * values. By making the number* B *of
bootstrap replications large, we allow the*
*random generation of bootstrap samples (sometimes called the Monte Carlo
method) to approximate as closely as we want the bootstrap distribution of *X** – . The histogram of bootstrap samples provides a replacement for
the sampling* *distribution of the *Z* or *t*
statistic used in confidence interval calculations. The histogram also replaces
the normal or *t* distribution tables
that we used in the para-metric approaches.

The idea behind the bootstrap is to approximate the
distribution of – *μ*. If this mimicking process
achieves that approximation, then we are able to draw infer-ences about *μ*. We have no particular reason to
believe that the mimicking process actually works.

The bootstrap statistical theory, developed since
1980, shows that under very general conditions, mimicking works as the sample
size *n* becomes large. Other
em-pirical evidence from simulation studies has shown that mimicking sometimes
works well even with small to moderate sample sizes (10–100). The procedure has
been modified and generalized to work for a wide variety of statistical
estimation problems.

The bootstrap principle is easy to remember and to
apply in general. You mimic the sampling from the population by sampling from
the empirical distribution. Wherever the unknown parameters appear in your
estimation formulae, you replace them by their estimates from the original
sample. Wherever the estimates appear in the formulae, you replace them with
their bootstrap estimates. The sample estimates and bootstrap estimates can be
thought of as actors. The sample estimates take on the role of the parameters
and the bootstrap estimates play the role of the sample es-timates.

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