#jsDisabledContent { display:none; } My Account |  Register |  Help

# Asymptotic distribution

Article Id: WHEBN0003001287
Reproduction Date:

 Title: Asymptotic distribution Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Asymptotic distribution

In mathematics and statistics, an asymptotic distribution is a distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical estimators.

## Definition

A sequence of distributions corresponds to a sequence of random variables Zi for i = 1, 2, ... . In the simplest case, an asymptotic distribution exists if the probability distribution of Zi converges to a probability distribution (the asymptotic distribution) as i increases: see convergence in distribution. A special case of an asymptotic distribution is when the sequence of random variables always approaches zero—that is, the Zi go to 0 as i goes to infinity. Here the asymptotic distribution is a degenerate distribution, corresponding to the value zero.

However, the most usual sense in which the term asymptotic distribution is used arises where the random variables Zi are modified by two sequences of non-random values. Thus if

Y_i=\frac{Z_i-a_i}{b_i}

converges in distribution to a non-degenerate distribution for two sequences {ai} and {bi} then Zi is said to have that distribution as its asymptotic distribution. If the distribution function of the asymptotic distribution is F then, for large n, the following approximations hold

P\left(\frac{Z_n-a_n}{b_n} \le x \right) \approx F(x) ,
P(Z_n \le z) \approx F\left(\frac{z-a_n}{b_n}\right) .

If an asymptotic distribution exists, it is not necessarily true that any one outcome of the sequence of random variables is a convergent sequence of numbers. It is the sequence of probability distributions that converges.

## The Central Limit Theorem

Perhaps the most common distribution to arise as an asymptotic distribution is the normal distribution. In particular, the central limit theorem provides an example where the asymptotic distribution is the normal distribution.

Central Limit Theorem
Suppose {X1, X2, ...} is a sequence of i.i.d. random variables with E[Xi] = µ and Var[Xi] = σ2 < ∞. Let Sn be the average of {X1, ..., Xn}. Then as n approaches infinity, the random variables n(Sn − µ) converge in distribution to a normal N(0, σ2):[1]

The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.

### Local Asymptotic normality

Local asymptotic normality is a generalization of the Central Limit Theorem. It is a property of a sequence of statistical models, which allows this sequence to be asymptotically approximated by a normal location model, after a rescaling of the parameter. An important example when the local asymptotic normality holds is in the case of iid sampling from a regular parametric model; this is just the Central Limit Theorem.

Barndorff-Nielson & Cox provide a direct definition of asymptotic normality.[2]

## References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.

By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.