# 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.

## 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, 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 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 → ∞.

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/n . 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/n . This approach is especially popular for the unit root tests.
• There are models where the dimension of the parameter space Θn slowly expands with n, reflecting the fact that the more observations there are, the more structural effects can be feasibly incorporated in the model.
• In kernel density estimation and kernel regression, an additional parameter is assumed—the bandwidth h. In those models, it is typically taken that h → 0 as n → ∞. The rate of convergence must be chosen carefully, though, usually hn−1/5.

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.

## 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:

${\displaystyle {\hat {\theta }}_{n}\ {\xrightarrow {\overset {}{p}}}\ \theta _{0}.}$

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.

#### 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

${\displaystyle b_{n}({\hat {\theta }}_{n}-a_{n})\ {\xrightarrow {d}}\ G,}$

then the sequence of estimators ${\displaystyle \textstyle {\hat {\theta }}_{n}}$ 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):

${\displaystyle {\sqrt {n}}({\hat {\theta }}_{n}-\theta _{0})\ {\xrightarrow {d}}\ {\mathcal {N}}(0,V).}$

## References

• Balakrishnan, N.; Ibragimov, I. A. V. B.; Nevzorov, V. B., eds. (2001), Asymptotic Methods in Probability and Statistics with Applications, Birkhäuser
• Borovkov, A. A.; Borovkov, K. A. (2010), Asymptotic Analysis of Random Walks, Cambridge University Press
• Buldygin, V. V.; Solntsev, S. (1997), Asymptotic Behaviour of Linearly Transformed Sums of Random Variables, Springer
• Le Cam, Lucien; Yang, Grace Lo (2000), Asymptotics in Statistics (2nd ed.), Springer
• DasGupta, A. (2008), Asymptotic Theory of Statistics and Probability, Springer
• Dawson, D.; Kulik, R.; Ould Haye, M.; Szyszkowicz, B.; Zhao, Y., eds. (2015), Asymptotic Laws and Methods in Stochastics, Springer-Verlag
• Höpfner, R. (2014), Asymptotic Statistics, Walter de Gruyter
• Lin'kov, Yu. N. (2001), Asymptotic Statistical Methods for Stochastic Processes, American Mathematical Society
• Oliveira, P. E. (2012), Asymptotics for Associated Random Variables, Springer
• Petrov, V. V. (1995), Limit Theorems of Probability Theory, Oxford University Press
• Sen, P. K.; Singer, J. M.; Pedroso de Lima, A. C. (2009), From Finite Sample to Asymptotic Methods in Statistics, Cambridge University Press
• Shiryaev, A. N.; Spokoiny, V. G. (2000), Statistical Experiments and Decisions: Asymptotic theory, World Scientific
• Small, C. G. (2010), Expansions and Asymptotics for Statistics, Chapman & Hall
• van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge University Press