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 constantly drawn from the sample data, which is considered as a

population, to generate the statistical deduction about original 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. According to our regression analysis lecture, bootstrapping is quite useful

in non-linear regression and generalized linear models. For small sample size,

the parametric bootstrapping method is highly preferred.2 In large

sample size, nonparametric bootstrapping method would be preferably utilized. For

a more detailed 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 one thousand or ten thousand

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 calculated 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)