# Asymptotic Normal Distribution

An asymptotic normal distribution is one that exhibits a property called asymptotic normality.

Asymptotic normality is a property of an estimator (like the sample mean or sample standard deviation). The term “Asymptotic” refers to how the estimator behaves as the sample size tends to infinity; an estimator that has an asymptotic normal distribution follow an approximately normal distribution as the sample size gets infinitely large.

An asymptotic normal distribution can be defined as the limiting distribution of a sequence of distributions. We’re often interested in the behavior of estimators as sample sizes get very large because estimators obtained from small samples are often biased (i.e., they deviate from the true population parameter you’re trying to estimate). When sample sizes get very large, the true population parameter (e.g., the population mean) and the estimator (e.g., the sample mean) will be equal and bias approaches zero. Under these circumstances, we can call the sample estimator a consistent estimator.

## Asymptotic Normal Distribution vs CLT

The property of asymptotic normality is like the Central Limit Theorem. The two concepts are so similar that in general terms, there really is no difference. However, the CLT is a theorem, and asymptotic normality is a property: one of weak convergence to a normal distribution. The property of asymptotic normality can be established with the CLT  .

## Formal Definition of Asymptotic Normality

An estimate has asymptotic normality if it converges on an unknown parameter at a “fast enough” rate, which Pachenko  defines as 1 / √(n). As an equation, an estimate has an asymptotic normal distribution if

holds true.

Sequences and probability distributions in general can also show asymptotic normality. For example, a sequence of random variables, dependent on a sample size n has an asymptotic normal distribution if two sequences μn and σn exist such that :

limn>∞ P[(Tn – μn) / σnx] = φ(x).

## References

 Lecture 4: Asymptotic Distribution Theory. Online: https://www.asc.ohio-state.edu/de-jong.8/note4.pdf

 Panchenko, D. (2006). Lecture 3 Properties of MLE: consistency, asymptotic normality, Fisher information. Online: https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2006/lecture-notes/lecture3.pdf

 Kolassa, J. (2014). Asymptotic Normality. DOI: https://doi.org/10.1007/978-3-642-04898-2_125

Content based on:
Stephanie Glen. “Asymptotic Normality” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/asymptotic-normality/