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.[1]

Overview

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 Xn, then the Xn converge in probability to the population mean E[Xi] as n → ∞.[2]

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.[2]

Besides the standard approach to asymptotics, other alternative approaches exist:

In many cases, highly accurate results for finite samples can be obtained via numerical methods (i.e. computers); even in such cases, though, asymptotic analysis can be useful. This point was made by Small (2010, §1.4), as follows.

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.

Modes of convergence of random variables

Further information: Convergence of random variables

Asymptotic properties

Estimators

Consistency

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 (the formula for generating the estimates) would almost surely give the correct result for the parameter being estimated.[2]

Asymptotic distribution

If it is possible to find sequences of non-random constants {an}, {bn} (possibly depending on the value of θ0), and a non-degenerate distribution G such that

then the sequence of estimators is said to have the asymptotic distribution G.

Most often, the estimators encountered in practice are asymptotically normal, meaning their asymptotic distribution is the normal distribution, with an = θ0, bn = n, and G = N(0, V):

Asymptotic confidence regions

Asymptotic theorems

See also

References

  1. ^ Höpfner, R. (2014), Asymptotic Statistics, Walter de Gruyter. 286 pag. ISBN 3110250241, ISBN 978-3110250244
  2. ^ a b c A.DasGupta. Asymptotic Theory of Statistics and Probability (2008) 756 pag. ISBN 0387759700, ISBN 978-0387759708

Bibliography