Does consistency imply the estimator is unbiased?

No. A consistent estimator can be biased in finite samples; consistency only requires that the bias and variance both vanish as the sample size grows, so the estimator concentrates on the true value in the limit.

What does the delta method do?

It gives the approximate distribution of a smooth function of an asymptotically normal estimator by linearizing the function, producing the function's value plus a normal error whose variance is scaled by the squared derivative.

Consistency and Asymptotic Normality

Consistency says an estimator homes in on the truth as data accumulate; asymptotic normality says its error, suitably scaled, becomes approximately normal, which is what makes standard errors meaningful.

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Definition

An estimator is consistent if it converges in probability to the true parameter as the sample size grows, and asymptotically normal if the rescaled estimation error converges in distribution to a normal law.

Scope

This topic covers convergence in probability and in distribution, the weak law of large numbers and the central limit theorem as the engines of consistency and asymptotic normality, the continuous mapping theorem and Slutsky's theorem, the delta method for the asymptotic distribution of smooth functions of an estimator, variance-stabilizing transformations, and the meaning of the resulting standard errors and confidence intervals.

Core questions

How do the law of large numbers and the central limit theorem yield consistency and asymptotic normality?
What do Slutsky's theorem and the continuous mapping theorem let you combine and transform?
How does the delta method give the asymptotic variance of a function of an estimator?
What is a variance-stabilizing transformation and why is it used?

Key theories

Consistency: By the law of large numbers and continuity arguments, well-behaved estimators converge in probability to the parameter they target, the minimal large-sample requirement for a sensible estimator.
Asymptotic normality and the delta method: The central limit theorem makes the scaled error of many estimators asymptotically normal, and the delta method transfers that normality, with a transformed variance, to smooth functions of the estimator.

Clinical relevance

Asymptotic normality is what licenses reporting an estimate with a standard error and a Wald confidence interval; the delta method in particular supplies standard errors for derived quantities such as odds ratios, ratios of means, and predicted probabilities throughout applied science.

History

The central limit theorem matured from Laplace through Lyapunov and Lindeberg in the early twentieth century. Cramer's 1946 treatise placed consistency, asymptotic normality, and the delta method at the center of mathematical statistics, where they remain.

Key figures

Pierre-Simon Laplace
Aleksandr Lyapunov
Harald Cramer
Aad van der Vaart

Seminal works

vanderVaart1998

Frequently asked questions

Does consistency imply the estimator is unbiased?: No. A consistent estimator can be biased in finite samples; consistency only requires that the bias and variance both vanish as the sample size grows, so the estimator concentrates on the true value in the limit.
What does the delta method do?: It gives the approximate distribution of a smooth function of an asymptotically normal estimator by linearizing the function, producing the function's value plus a normal error whose variance is scaled by the squared derivative.