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There are some duplicates since a bootstrap resample comes from sampling with replacement from the data. Also the number of data points in a bootstrap resample is equal to the number of data points in our original observations. Then we compute the mean of this resample and obtain the first bootstrap mean: μ 1 *.
Sampling done without replacement is no longer independent, but still satisfies exchangeability, hence most results of mathematical statistics still hold. Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the probability of choosing the same individual ...
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 ...
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 ...
Sampling with replacement ensures each bootstrap is independent from its peers, as it does not depend on previous chosen samples when sampling. Then, m {\displaystyle m} models are fitted using the above bootstrap samples and combined by averaging the output (for regression) or voting (for classification).
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one.
In all the injective cases (sampling without replacement), the number of sets of choices is zero unless N ≤ X. ("Comparable" in the above cases means that each element of the sample space of the corresponding distribution corresponds to a separate set of choices, and hence the number in the appropriate box indicates the size of the sample ...
Suppose marbles are drawn without replacement until the urn is empty. Let be the indicator random variable of the event that the -th marble drawn is red. Then {} =, …, + is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.