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 typically assumed that the sample size n grows indefinitely; the properties of estimators and tests are then evaluated in the limit as n → ∞. In practice, a limit evaluation is treated as being approximately valid for large finite sample sizes, as well.A primary goal of asymptotic analysis is to obtain a deeper qualitative understanding of quantitative tools. The conclusions of an asymptotic analysis often supplement the conclusions which can be obtained by numerical methods. 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 n grows indefinitely; the properties of estimators and tests are then evaluated in the limit as n → ∞. In practice, a limit evaluation is treated as being approximately valid for large finite sample sizes, as well. 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, so 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 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 Xn, then the Xn converge in probability to the population mean E 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.