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| In [[mathematics]] and [[statistics]], an '''asymptotic distribution''' is a hypothetical 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 function]]s of statistical [[estimator]]s.
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| ==Definition==
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| A sequence of distributions corresponds to a sequence of random variables Z<sub>i</sub> for ''i'' = 1, 2, ... . In the simplest case, an asymptotic distribution exists if the probability distribution of Z<sub>i</sub> converges to a probability distribution (the asymptotic distribution) as ''i'' increases: see [[Convergence of random variables#Convergence in distribution|convergence in distribution]]. A special case of an asymptotic distribution is when the sequence of random variables always approaches zero—that is, the Z<sub>i</sub> go to 0 as i goes to infinity. Here the asymptotic distribution is a [[degenerate distribution]], corresponding to the value zero.
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| However, the most usual sense in which the term asymptotic distribution is used arises where the random variables Z<sub>i</sub> are modified by two sequences of non-random values. Thus if
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| :<math>Y_i=\frac{Z_i-a_i}{b_i}</math> | |
| converges in distribution to a non-degenerate distribution for two sequences {''a<sub>i</sub>''} and {''b<sub>i</sub>''} then Z<sub>i</sub> 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
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| :<math>P\left(\frac{Z_n-a_n}{b_n} \le x \right) \approx F(x) ,</math>
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| :<math>P(Z_n \le z) \approx F\left(\frac{z-a_n}{b_n}\right) .</math>
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| 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.
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| ==Asymptotic normality==
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| {{seealso|local asymptotic normality}}
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| 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]].
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| Barndorff-Nielson & Cox<ref>Barndorff-Nielson, O.E., Cox, D.R. (1989) Asymptotic Techniques for Use in Statistics. Chapman and Hall. ISBN 0-412-31400-2</ref> provide a direct definition of asymptotic normality.
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| The [[Binomial distribution]] seems to be the first asymptotic approximation by a [[normal distribution]]; compared to the more general case of [[central limit theorem]], convergence of the Binomial to the normal is especially rapid.{{cn|date=March 2011}} Asymptotic normality of the [[Binomial distribution]] is proven by the [[de Moivre–Laplace theorem]].{{cn|date=March 2011}}
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| ==See also==
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| *[[Asymptotic theory]]
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| *[[Central limit theorem]]
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| *[[de Moivre–Laplace theorem]]
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| *[[Limiting density of discrete points]]
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| ==References==
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| {{Reflist}}
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| [[Category:Theory of probability distributions]]
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| [[Category:Asymptotic statistical theory]]
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Myrtle Benny is how I'm known as and I feel comfy when people use the complete title. I am a meter reader. He is truly fond of doing ceramics but he is struggling to find time for it. South Dakota is where me and my husband reside.
Feel free to surf to my web page ... exchange.craftemergency.org