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The best example of the plug-in principle, the bootstrapping method. Bootstrapping is a statistical method for estimating the sampling distribution of an estimator by sampling with replacement from the original sample, most often with the purpose of deriving robust estimates of standard errors and confidence intervals of a population parameter like a mean, median, proportion, odds ratio ...
Although simple random sampling can be conducted with replacement instead, this is less common and would normally be described more fully as simple random sampling with replacement. Sampling done without replacement is no longer independent, but still satisfies exchangeability , hence most results of mathematical statistics still hold.
The bootstrap sample is taken from the original by using sampling with replacement (e.g. we might 'resample' 5 times from [1,2,3,4,5] and get [2,5,4,4,1]), so, assuming N is sufficiently large, for all practical purposes there is virtually zero probability that it will be identical to the original "real" sample. This process is repeated a large ...
Sampling schemes may be without replacement ('WOR' – no element can be selected more than once in the same sample) or with replacement ('WR' – an element may appear multiple times in the one sample). For example, if we catch fish, measure them, and immediately return them to the water before continuing with the sample, this is a WR design ...
"Adjusted jackknife for imputation under unequal probability sampling without replacement". Journal of the Royal Statistical Society, Series B. 68 (3): 531– 547. doi: 10.1111/j.1467-9868.2006.00555.x. Berger, Y.G.; Skinner, C.J. (2005). "A jackknife variance estimator for unequal probability sampling". Journal of the Royal Statistical Society ...
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Sampling without replacement where order does matter does not seem to correspond to a probability distribution. [2] In all the injective cases (sampling without replacement), the number of sets of choices is zero unless N ≤ X.
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