Sampling with replacement

Finally, sampling with replacement means that in theory, after each portion is selected and measured, it is returned to the total sample pool and thus has the opportunity to be selected again. This is a corollary of the assumption of independence. Violation of this assumption (which is almost always the case in toxicology and all the life sciences) does not have serious consequences if the total pool from which samples are sufficiently large (say 20 or greater) so that the chance of reselecting that portion is small anyway. 

Characters are randomly sampled with replacement, leading to 13,14 new data set of same size as original... 

Bootstrapping involves the repetitive drawing of random samples with replacement from the observed population and computing statistics. A complete bootstrap in an observed population with eight variables would require the calculation of bootstrap statistics for 8 = 16,777,216 samples, quite a computer-intensive process. Therefore bootstrap samples are usually Umited to hundreds or thousands of drawings. 

A.4.2 This is a case of sampling with replacement. There are 150 successive boxes or slots, each of which can take one of four possible values or states, thus = 4150 = 2 037035976334487 x 10 . 

A bootstrap sample is generated by repeated random sampling, with replacement, of an m-sized pseudosample from the original data set. At each sampling step, every vector x, has an equal probability of being chosen. Thus, for a given iteration, it is possible to choose three of Xi, none of X2, five of X3, and so forth. 

The Cl is [-0.144, -0.108] and does not contain zero, supporting the notion that the two elimination rate constants do differ. An alternative approach to the above would be to replace the Wald based confidence intervals with those produced using the nonparametric bootstrap technique. With this technique the data set is sampled with replacement at the subject level many times, and the model is fit to each of these resampled data sets, generating an empirical distribution for each model parameter. Confidence intervals can then be constructed for the model parameters based on the percentiles of their empirical distributions. 

One method to estimate standard errors is the non-parametric bootstrap (see the book Appendix for further details and background). With this method, subjects are repeatedly sampled with replacement creating a new data set of the same size as the original dataset. For example, if the data set had 100 subjects with subjects numbered 1,2,..., 100. The first bootstrap data set may... 

The bootstrap cannot just be applied to any problem. There do exist situations where the bootstrap will fail, particularly those situations where the statistic depends on a very narrow feature of the original sampling process (Stine, 1990), such as the sample maximum. For instance, the bootstrap may have poor coverage in estimating the Cl associated with maximal drug concentrations (Cmar). The bootstrap will also have trouble when the sample size is small although what constitutes small is debatable. The number of possible combinations drawn from an n-size sample with replacement is... 

Divide data into k subsamples. Each subsample is a simple random sample with replacement. Compare performance of k subsamples. 

This procedure is known as sampling with replacement. If we had to subject the sample to destructive assays, as is sometimes the case with routine inspection of factory production lines, obviously there would be no replacement. 

In bootstrapping, we repeatedly analyze subsamples, instead of subsets of the known set. Each subsample is a random sample with replacement from the full sample (known set). Bootstrapping seems to perform better than cross-validation in many instances (Efron, 1983). 

F -) in place of the true c.d.f. Ff). We will now consider the population to be the observed data having c.d.f. which places mass 1 jn on each of the observed data values X,. Thus, we select M random samples of size n (sampling with replacement) from this new population and compute 0i, i,.. .,6m- We now have M realizations of d from which we can estimate the p.d.f. (using a kernel density estimator), the quantile function, or specific parameters like its mean. 

Resampling methods draw repeated samples from the observed sample itself to generate the sampling distribution of a statistic. The permutation method draws samples without replacement while the bootstrap method draws samples with replacement. These methods are useful for assessing the accuracy (e.g., bias and standard error) of complex statistics. 

The bagging algorithm uses bootstrap samples to build base classifiers. Each bootstrap sample is formed by randomly sampling, with replacement, the same number of observations as the training set. The final classification produced by the ensemble of these base classifiers is obtained using equal-weight voting. 

This is an example of sampling with replacement. Suppose a box contains b blue marbles and r red marbles. Let us perform n trials of an experiment in which a marble is chosen at random, its color is observed, and the marble is put back in the box. 

We start with the measured AMetric for each sample at a given temperature and then perform uniform random sampling with replacement of this data to generate another set with the same number of samples. This process is repeated for each... 

It is noteworthy that in this quantification it has been assumed Poisson sampling with replacement (see Eq. 8). This approximation, however, is very... 

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