Asymptotic theory (statistics)

Template:Short descriptionIn 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 Template:Math grows indefinitely; the properties of estimators and tests are then evaluated in the limit as Template:Math. 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 Template:Math. 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. Template:Math. 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 Template:Math, if one value is drawn from each random variable and the average of the first Template:Math values is computed as Template:Math, then the Template:Math converge in probability to the population mean Template:Math as Template:Math.

In asymptotic theory, the standard approach is Template:Math. 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: Template:Math and Template:Math, 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 Template:Math, such that the Template:Math-th model corresponds to Template:Math. 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 Template:Math and the alternative is Template:Math. This approach is especially popular for the unit root tests.
• There are models where the dimension of the parameter space Template:Math slowly expands with Template:Math, 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 Template:Math. In those models, it is typically taken that Template:Math as Template:Math. The rate of convergence must be chosen carefully, though, usually Template:Math.

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 Template:Harvtxt, as follows. Template:Quote

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:

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

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 Template:Math, Template:Math (possibly depending on the value of Template:Math), and a non-degenerate distribution Template:Math such that

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

then the sequence of estimators $\textstyle {\hat {\theta }}_{n}$ 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 Template:Math, Template:Math, and Template:Math:

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

Asymptotic theorems

• Central limit theorem
• Continuous mapping theorem
• Glivenko–Cantelli theorem
• Law of large numbers
• Law of the iterated logarithm
• Slutsky’s theorem

See also

• Asymptotic analysis
• Exact statistics
• Large deviations theory

References

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