In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is typically assumed that the sample size grows indefinitely; the properties of estimators and tests are then evaluated in the limit as. In practice, a limit evaluation is treated as being approximately valid for large finite sample sizes, as well.
Overview
Most statistical problems begin with a dataset of size. The asymptotic theory proceeds by assuming that it is possible to keep collecting additional data, so that the sample size grows infinitely, i.e.. Under the assumption, many results can be obtained that are unavailable for samples of finite size. An example is the weak law of large numbers. The law states that for a sequence of independent and identically distributed random variables, if one value is drawn from each random variable and the average of the first values is computed as, then the converge in probability to the population mean as. In asymptotic theory, the standard approach is. For some statistical models, slightly different approaches of asymptotics may be used. For example, with panel data, it is commonly assumed that one dimension in the data remains fixed, whereas the other dimension grows: and, or vice versa. Besides the standard approach to asymptotics, other alternative approaches exist:
Within the local asymptotic normality framework, it is assumed that the value of the "true parameter" in the model varies slightly with, such that the -th model corresponds to. This approach lets us study the regularity of estimators.
When statistical tests are studied for their power to distinguish against the alternatives that are close to the null hypothesis, it is done within the so-called "local alternatives" framework: the null hypothesis is and the alternative is. This approach is especially popular for the unit root tests.
There are models where the dimension of the parameter space slowly expands with, reflecting the fact that the more observations there are, the more structural effects can be feasibly incorporated in the model.
In many cases, highly accurate results for finite samples can be obtained via numerical methods ; even in such cases, though, asymptotic analysis can be useful. This point was made by, as follows.
A sequence of estimates is said to be consistent, if it converges in probability to the true value of the parameter being estimated: That is, roughly speaking with an infinite amount of data the estimator would almost surely give the correct result for the parameter being estimated.
If it is possible to find sequences of non-random constants, , and a non-degenerate distribution such that then the sequence of estimators is said to have the asymptotic distributionG. Most often, the estimators encountered in practice are asymptotically normal, meaning their asymptotic distribution is the normal distribution, with,, and :
