Asymptotic theory (statistics)

In statistics, asymptotic theory, or large sample theory, is a generic framework for assessment of properties of estimators and statistical tests. Within this framework it is typically assumed that the sample size n grows indefinitely, and the properties of statistical procedures are evaluated in the limit as n → ∞.

In practical applications, asymptotic theory is applied by treating the asymptotic results as approximately valid for finite sample sizes as well. Such approach is often criticized for not having any mathematical grounds behind it, yet it is used ubiquitously anyway. The importance of the asymptotic theory is that it often makes possible to carry out the analysis and state many results which cannot be obtained within the standard “finite-sample theory”.

Overview

Most statistical problems begin with a dataset of size n. The asymptotic theory proceeds by assuming that it is possible to keep collecting additional data, so that the sample size would grow infinitely:

n\to \infty \,

Under this assumption many results can be obtained that are unavailable for samples of finite sizes. As an example consider the law of large numbers. This law states that for a sequence of iid random variables X₁, X₂, …, the sample averages $\scriptstyle {\overline {X}}_{n}$ converge in probability to the population mean E[X_i] as n → ∞. At the same time for finite n it is impossible to claim anything about the distribution of $\scriptstyle {\overline {X}}_{n}$ if the distributions of individual X_i’s is unknown.

For various models slightly different modes of asymptotics may be used:

For cross-sectional data (iid) the new observations are sampled independently, from the same fixed distribution. This is the standard case of n → ∞ asymptotics.
For longitudinal data (time series) the sampling method may differ from model to model. Sometimes the data is assumed to be ergodic, in other applications it can be integrated or cointegrated. In this case the asymptotic is again taken as the number of observations (usually denoted T in this case) goes to infinity: T → ∞.
For panel data, it is commonly assumed that one dimension in the data (T) remains fixed, whereas the other dimension grows: T = const, n → ∞.

Besides these standard approaches, various other “alternative” asymptotic 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 $\scriptstyle \theta _{n}\,=\,\theta +h/{\sqrt {n}}$ . 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: $\scriptstyle \theta \,=\,\theta _{0}+h/{\sqrt {n}}$ . 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 a statistician has, the more he is tempted to introduce additional parameters in the model. An example of this is the weak instruments asymptotic.
In kernel density estimation and kernel regression additional parameter — the bandwidth h — is assumed. In these models it is typically taken that h → 0 as n → ∞, however the rate of convergence must be chosen carefully, usually h ∝ n^−1/5.

Modes of convergence of random variables

Further information: Convergence of random variables

Asymptotic properties

Estimators

Consistency: a sequence of estimators 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}

Generally an estimator is just some, more or less arbitrary, function of the data. The property of consistency requires that the estimator was estimating the quantity we intended it to. As such, it is the most important property in the estimation theory: estimators that are known to be inconsistent are never used in practice.

Asymptotic distribution: if it is possible to find sequences of non-random constants {a_n}, {b_n} (possibly depending on the value of θ₀), and a non-degenerate distribution G such that

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

then the sequence of estimators $\scriptstyle\hat\theta_n$ is said to have the asymptotic distribution G.

Most often, the estimators encountered in practice have the asymptotically normal distribution, with $a n = θ 0$ , $b n = \sqrt n$ , and $G = N (0, V)$ :

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

Asymptotic confidence regions.
Regularity.

Asymptotic theorems

Notes

References

Le Cam, Lucien; Yang, Grace Lo (2000). Asymptotics in statistics (some basic concepts) (2nd ed.). Springer. ISBN 0-387-95036-2.
van der Vaart, A.W. (1998). Asymptotic statistics. Cambridge University Press. ISBN 978-0-521-49603-2.

Statistics

Descriptive statistics

Continuous data

Center	Mean arithmetic geometric harmonic Median Mode

Dispersion	Variance Standard deviation Coefficient of variation Percentile Range Interquartile range

Shape	Moments Skewness Kurtosis L-moments

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Population Statistic Effect size Statistical power Sample size determination Missing data

Survey methodology	Sampling Standard error stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in

Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife

Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F

Goodness of fit	Chi-squared Kolmogorov–Smirnov Anderson–Darling Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC

Rank statistics	Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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