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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 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 ...
(The sample mean need not be a consistent estimator for any population mean, because no mean needs to exist for a heavy-tailed distribution.) A well-defined and robust statistic for the central tendency is the sample median, which is consistent and median-unbiased for the population median.
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 ...
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 ...
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.
To then oversample, take a sample from the dataset, and consider its k nearest neighbors (in feature space). To create a synthetic data point, take the vector between one of those k neighbors, and the current data point. Multiply this vector by a random number x which lies between 0, and 1. Add this to the current data point to create the new ...
Let a be the value of our statistic as calculated from the full sample; let a i (i = 1,...,n) be the corresponding statistics calculated for the half-samples. (n is the number of half-samples.) Then our estimate for the sampling variance of the statistic is the average of (a i − a) 2. This is (at least in the ideal case) an unbiased estimate ...