Instatistics,asymptotic theory, orlarge sample theory, is a framework for assessing properties ofestimators andstatistical tests. Within this framework, it is often assumed that thesample sizen may grow indefinitely; the properties of estimators and tests are then evaluated under the limit ofn → ∞. In practice, a limit evaluation is considered to be approximately valid for large finite sample sizes too.^[1]

Most statistical problems begin with a dataset ofsizen. 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 theweak law of large numbers. The law states that for a sequence ofindependent and identically distributed (IID)random variablesX₁,X₂, ..., if one value is drawn from each random variable and the average of the firstn values is computed asX_n, then theX_nconverge in probability to the population meanE[X_i] asn → ∞.^[2]

In asymptotic theory, the standard approach isn → ∞. For somestatistical models, slightly different approaches of asymptotics may be used. For example, withpanel data, it is commonly assumed that one dimension in the data remains fixed, whereas the other dimension grows:T = constant andN → ∞, or vice versa.^[2]

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

• Within thelocal asymptotic normality framework, it is assumed that the value of the "true parameter" in the model varies slightly withn, such that then-th model corresponds toθ_n =θ +h/√n. This approach lets us study theregularity of estimators.
• Whenstatistical 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 isH₀:θ =θ₀ and the alternative isH₁:θ =θ₀ +h/√n. This approach is especially popular for theunit root tests.
• There are models where the dimension of the parameter spaceΘ_n slowly expands withn, reflecting the fact that the more observations there are, the more structural effects can be feasibly incorporated in the model.
• Inkernel density estimation andkernel regression, an additional parameter is assumed—the bandwidthh. In those models, it is typically taken thath → 0 asn → ∞. The rate of convergence must be chosen carefully, though, usuallyh ∝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 bySmall (2010, §1.4), as follows.

A primary goal of asymptotic analysis is to obtain a deeperqualitative understanding ofquantitative tools. The conclusions of an asymptotic analysis often supplement the conclusions which can be obtained by numerical methods.

A sequence of estimates is said to beconsistent, if itconverges in probability to the true value of the parameter being estimated:

${\displaystyle {\hat{\theta }}_n\stackrel{\stackrel{}{p}}{\to }{\theta }_0.}$

That is, roughly speaking with an infinite amount of data theestimator (the formula for generating the estimates) would almost surely give the correct result for the parameter being estimated.^[2]

If it is possible to find sequences of non-random constants{a_n},{b_n} (possibly depending on the value ofθ₀), and a non-degenerate distributionG such that

${\displaystyle b_n({\hat{\theta }}_n-a_n)\stackrel{d}{\to }G,}$

then the sequence of estimators${\displaystyle {\hat{\theta }}_n}$ is said to have theasymptotic distributionG.

Most often, the estimators encountered in practice areasymptotically normal, meaning their asymptotic distribution is thenormal distribution, witha_n =θ₀,b_n =√n, andG =N(0,V):

${\displaystyle \sqrt{n}({\hat{\theta }}_n-{\theta }_0)\stackrel{d}{\to }\mathcal{N}(0,V).}$

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

• Asymptotic analysis
• Exact statistics
• Large deviations theory

• Asymptotic theory (statistics)

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