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In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size n may grow indefinitely; the properties of estimators and tests are then evaluated under the limit of n → ∞. In practice, a limit evaluation is ...
The continuous mapping theorem states that for a continuous function g, if the sequence {X n} converges in distribution to X, then {g(X n)} converges in distribution to g(X). Note however that convergence in distribution of {X n} to X and {Y n} to Y does in general not imply convergence in distribution of {X n + Y n} to X + Y or of {X n Y n} to XY.
In statistics, an empirical distribution function (commonly also called an empirical cumulative distribution function, eCDF) is the distribution function associated with the empirical measure of a sample. [1] This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. Its value at any specified ...
The simplest bootstrap method involves taking the original data set of heights, and, using a computer, sampling from it to form a new sample (called a 'resample' or bootstrap sample) that is also of size N. 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 ...
When g is applied to a random variable such as the mean, the delta method would tend to work better as the sample size increases, since it would help reduce the variance, and thus the taylor approximation would be applied to a smaller range of the function g at the point of interest.
In the case of independent samples, the convergence rate is n −1/2, where n is the sample size, and the constant is estimated in terms of the third absolute normalized moment. It is also possible to give non-uniform bounds which become more strict for more extreme events.
For example, for an iid sample {x 1,..., x n} one can use T n (X) = x n as the estimator of the mean E[X]. Note that here the sampling distribution of T n is the same as the underlying distribution (for any n, as it ignores all points but the last), so E[T n (X)] = E[X] and it is unbiased, but it does not converge to any value.