Asymptotic theory (statistics)

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 considered to be approximately valid for large finite sample sizes too. Most statistical problems begin with a dataset of size n. The asymptotic theory proceeds by assuming that it is possible (in principle) to keep collecting additional data, thus that the sample size grows infinitely, i.e. n → ∞. 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 (IID) random variables X1, X2, ..., if one value is drawn from each random variable and the average of the first n values is computed as n, then the n converge in probability to the population mean E[Xi] as n → ∞. In asymptotic theory, the standard approach is n → ∞. 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: T = constant and N → ∞, 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 n, such that the n-th model corresponds to θn = θ + h/ . 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 H0: θ = θ0 and the alternative is H1: θ = θ0 + h/ . This approach is especially popular for the unit root tests.