largesample

Large-sample theory, in statistics, concerns the behavior of estimators and test statistics as the sample size n grows without bound. It provides asymptotic approximations that justify inference when exact finite-sample results are unavailable or intractable.

Two foundational results are central to large-sample reasoning: the law of large numbers, which states that sample means converge to the population mean as n tends to infinity, and the central limit theorem, which implies that suitably normalized sums converge in distribution to a normal distribution. These results underpin approximate confidence intervals and hypothesis tests for large samples.

Under regularity conditions, estimators such as maximum likelihood estimators are consistent and asymptotically normal, with variance given by the inverse Fisher information. The delta method extends these results to smooth functions of parameters. Hypothesis tests and confidence intervals often rely on Wald, score, or likelihood ratio statistics that have approximate chi-square or normal distributions in large samples.

Practical use of large-sample theory includes guiding study design and the interpretation of results when data sets are large. It contrasts with finite-sample methods that provide exact results for small samples. When regularity fails or sample sizes are moderate, resampling and simulation, such as bootstrap methods, offer alternatives to rely on asymptotic approximations.

See also: asymptotic theory, law of large numbers, central limit theorem, maximum likelihood estimation, delta method, bootstrap.