In regression analysis, bootstrapping is an efficient tool for statistical

deduction, which focused on making a sampling distribution with the key idea of

resampling the originally observed data with replacement1. 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’2. So, from the meaning

of this concept, sample data is considered as a population and replacement

samples are repeatedly drawn from the sample data, which is considered as a

population, to generate the statistical deduction about the sample data. The essential bootstrap analogy states that “the

population is to the sample as the sample is to the bootstrap samples”2.

The bootstrap falls into two types, parametric and nonparametric. Parametric

bootstrapping assumes that the original data set is drawn from some specific

distributions, e.g. normal distribution2. And the samples generally are

pulled as the same size as the original data set. Nonparametric

bootstrapping is right the one described in the start of this summary, which repeatedly

and randomly draws a certain size of 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).2 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 necessary. 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 * note is added to distinguish resampled data from original data. Replacement

is mandatory and supposed to be repeated typically thousands of times, which is

still developing since computation power develops, otherwise only original

sample A would be generated.1 And for each bootstrap estimate of these samples, mean

is calculated to estimate the expectation of the bootstrapped statistics. Mean minus 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)