Bootstrapping is a method for statistical inference, which focused on

building a sampling distribution with the key idea of resampling the originally

observed data with replacement. The term bootstrapping, proposed by Bradley

Efron in his “Bootstrap methods: another look at the jackknife” published

in 1979, is extracted from the cliché of ‘pulling oneself up by one’s

bootstraps’. So, from the meaning of this concept, sample data is considered as

a population and repeated samples are drawn from the sample data, which is

considered as a population, to generate the statistical inference about the sample

data. The essential bootstrap analogy states

that “the population is to the sample as the sample is to the bootstrap samples”.

The bootstrap was divided into two methods, which are parametric and

nonparametric. Parametric bootstrapping assumes that the original data set is drawn

from some specific distributions, e.g. normal distribution. And the sample often has the

same size as the original data set.

Nonparametric bootstrapping is just the

one described in the beginning, which draws bootstrapping samples from the

original data. Bootstrapping is quite useful in non-linear regression and

generalized linear models. For small sample size, the parametric bootstrapping

method is highly preferred. In large sample size, nonparametric bootstrapping

method would be preferably utilized. For a further clarification of nonparametric

bootstrapping, a sample data set, A = {x1, x2, …, xk} is randomly drawn from

a population B = {X1, X2, …, XK} and K is much larger than k. The statistic T

= t(A) is considered as an estimate of the corresponding population parameter P

= t(B). Nonparametric bootstrapping generates the estimate of the sampling

distribution of a statistic in an empirical way. No assumptions of the form of the population

is needed. Next, a sample of size k is drawn from the elements of A with replacement,

which represents as A?1 =

{x?11,

x?12,

…, x?1k}.

In the resampling, a * is added to distinguish re-sampled data from original

data. Replacement is mandatory and supposed to be repeated typically 1000 or

10000 times, which is still increasing since computation power increases, otherwise

only original sample A would be generated. And for each bootstrap estimate of these samples, mean is

calculated to estimate the expectation of the bootstrapped statistics. Mean-T is the estimate of T’s bias. And T?, the bootstrap variance estimate, estimates the sampling variance of the population, P. Then

bootstrap confidence intervals can be constructed using either bootstrap

percentile interval approach or normal theory interval approach. Confidence

intervals by bootstrap percentile method is to use the empirical quantiles of the

bootstrap estimates, which is written as T?(lower) < P < T?(upper). More
specifically, it can be written as Tˆ ? (Tˆ ? upper – T*ˆ) ? P ? Tˆ + (T*ˆ + Tˆ ?lower).
Bootstrapping is an effective
method to doublecheck the stability of the model estimation results. It is much
better than the intervals calculated by sample variance with normality
assumption. And simplicity is bootstrapping's another important benefit. For
complicated estimators, such as correlation coefficients, percentile points,
for complex parameters in the distribution, it is a pretty simple way to generate
estimates of confidence intervals and standard errors. However, simplicity can also
bring up disadvantage for bootstrapping, which makes the important assumptions for
the bootstrapping easy to neglect. And bootstrapping is often over-optimistic and
doesn't assure finite sample size.
There are several types of bootstrapping schemes in the regression
problems. One typical approach is to resample residuals in the regression models.
The main procedure is firstly fit the original data set with the model, and generate
model estimates, ?ˆ and calculate residuals, ?ˆ; secondly randomly and repeatedly
sample the residuals (typically 1000 or 10000 times) to get K sets residuals of
size k and add each resampled residual to the original equation, generating
bootstrapped Y*; Finally use bootstrapped Y* to refit the model and get bootstrap
estimate ?ˆ?.
Another typical approach in the regression context is random-x resampling,
which is also called case resampling. We can either apply Monte Carlo algorithm,
which is to repeatedly resample the data of the same size as the original data
set with replacement, or identify any possible resampling of the data set. In
our case, before fitting regression model with the original predictor variable
and response pairs (xi, yi), for i = 1, 2, . . ., k, these data pairs are resampled
to get K new data pairs of size k. Then the regression model is fit to each of
these K new data sets. ?ˆ? is
generated from K parameter estimates.