In regression analysis, 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 falls into two types, parametric and nonparametric. Parametric
bootstrapping assumes that the original data set is drawn from some specific
distributions, e.g. normal distribution. And the samples generally are pulled as the same size
as the original data set. Nonparametric bootstrapping is just the one described
in the beginning, which draws a portion 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). 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
1000 or 10000 times, which is still developing since computation power develops,
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 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)

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