Why rely on asymptotics rather than exact distributions?

Exact finite-sample distributions are usually unknown or intractable, whereas limiting normal and chi-squared approximations are simple, broadly applicable, and accurate for moderate sample sizes.

How large must the sample be for asymptotics to apply?

There is no universal answer; it depends on the model, the parameter, and the skewness of the data. The approximations can be excellent for a few dozen observations or poor for hundreds near a boundary, which is why resampling checks are common.

Asymptotic Theory

Asymptotic theory studies how estimators and tests behave as the sample size grows without bound, providing tractable approximations when exact distributions are intractable.

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Definition

Asymptotic theory is the part of mathematical statistics that derives limiting distributions and approximations for statistical procedures as the sample size tends to infinity, and uses them to compare and justify those procedures.

Scope

This area covers modes of convergence and the continuous mapping and Slutsky theorems, consistency of estimators, asymptotic normality and the delta method, M- and Z-estimation as a unifying framework for estimators defined by maximization or estimating equations, empirical-process theory and uniform laws and central limit theorems over function classes, contiguity, local asymptotic normality, and the convolution and local-asymptotic-minimax theorems that define asymptotic efficiency.

Sub-topics

Core questions

What does it mean for an estimator to be consistent and asymptotically normal?
How does the delta method propagate asymptotic normality through smooth transformations?
How does M-estimation unify maximum likelihood, least squares, and robust estimators?
What is asymptotic efficiency, and how does Le Cam's theory characterize the best limiting variance?

Key theories

Consistency and asymptotic normality: Under regularity, estimators converge in probability to the true parameter and, rescaled by the square root of the sample size, converge to a normal distribution, justifying standard errors and Wald confidence intervals.
M-estimation and empirical processes: Estimators maximizing a sample criterion or solving estimating equations are analyzed uniformly via empirical-process theory, which supplies the uniform laws of large numbers and central limit theorems the arguments require.
Local asymptotic normality and efficiency: Le Cam's local asymptotic normality reduces a smooth model near the truth to a normal experiment; the convolution and local-asymptotic-minimax theorems then define the best achievable asymptotic variance.

Clinical relevance

Asymptotic approximations supply the standard errors, Wald and likelihood-ratio confidence intervals, and large-sample tests reported by essentially all statistical software, so the validity of routine inference in the sciences rests on these limit theorems holding to good approximation.

History

Building on the classical central limit theorem, Le Cam developed the theory of contiguity, local asymptotic normality, and asymptotic efficiency from the 1950s onward. Hajek's convolution theorem and the empirical-process program of the late twentieth century, synthesized by van der Vaart, completed the modern framework.

Key figures

Lucien Le Cam
Aad van der Vaart
Jaroslav Hajek
Peter J. Bickel

Seminal works

vanderVaart1998

Frequently asked questions

Why rely on asymptotics rather than exact distributions?: Exact finite-sample distributions are usually unknown or intractable, whereas limiting normal and chi-squared approximations are simple, broadly applicable, and accurate for moderate sample sizes.
How large must the sample be for asymptotics to apply?: There is no universal answer; it depends on the model, the parameter, and the skewness of the data. The approximations can be excellent for a few dozen observations or poor for hundreds near a boundary, which is why resampling checks are common.