An asymptotic test is one that uses an estimator that converges to a limit under very general conditions as the sample size increases. There are a number of kinds of convergence in this context: convergence in distribution (the weakest form of convergence), convergence in probability (also known as the weak law of large numbers), almost sure convergence (also known as point convergence, or the strong law of large numbers).
For example, if n, the sample size, increases toward infinity, the estimator's distribution converges toward the theoretical distribution (which might be, for example, the normal, chi-square, or F). Or, the mean of the estimator may converge toward the mean of the theoretical distribution.
An asymptotic test is one that uses an estimator that converges to a limit under very general conditions as the sample size increases. There are a number of kinds of convergence in this context: convergence in distribution (the weakest form of convergence), convergence in probability (also known as the weak law of large numbers), almost sure convergence (also known as point convergence, or the strong law of large numbers).
For example, if n, the sample size, increases toward infinity, the estimator's distribution converges toward the theoretical distribution (which might be, for example, the normal, chi-square, or F). Or, the mean of the estimator may converge toward the mean of the theoretical distribution.
You can use an asymptotic test if you have enough sample size, otherwise an exact test or a corrected asymptotic test should be used.
Asymptotic tests are valid under the assumption that the sample size n is large that is n → Infinity.
In practical applications, asymptotic theory is applied by treating the asymptotic results as approximately valid for finite sample sizes as well. Such approach is often criticized for not having any mathematical grounds behind it, yet it is used ubiquitously anyway. The importance of the asymptotic theory is that it often makes possible to carry out the analysis and state many results which cannot be obtained within the standard “finite-sample theory”.
n → ∞.
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