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.

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 $X_{1}, X_{2}, ...$, if one value is drawn from each random variable and the average of the first $n$ values is computed as $\overline{X}_{n}$, then the $\overline{X}_{n}$ converge in probability to the population mean $E[X_{i}]$ 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 $θ_{n} = θ + h/√n$. 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 $H_{0}: θ = θ_{0}$ and the alternative is $H_{1}: θ = θ_{0} + h/√n$. This approach is especially popular for the unit root tests.
 * There are models where the dimension of the parameter space $Θ_{n}$ 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 $h ∝ n^{−1/5}$.

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, 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."

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 ${a_{n}}$, ${b_{n}}$ (possibly depending on the value of $θ_{0}$), and a non-degenerate distribution $G$ 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 $a_{n} = θ_{0}$, $b_{n} = √n$, and $G = N(0, V)$:
 * $$\sqrt{n}(\hat\theta_n - \theta_0)\ \xrightarrow{d}\ \mathcal{N}(0, V).$$

Asymptotic theorems

 * Central limit theorem
 * Continuous mapping theorem
 * Glivenko–Cantelli theorem
 * Law of large numbers
 * Law of the iterated logarithm
 * Slutsky's theorem
 * Delta method