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In [[statistics]], '''asymptotic theory''', or '''large sample theory''', is a generic framework for assessment of properties of [[estimator]]s and [[statistical test]]s. Within this framework it is typically assumed that the [[sample size]] ''n'' grows indefinitely, and the properties of statistical procedures are evaluated in the limit as {{nowrap|''n'' → ∞}}.
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In practical applications, asymptotic theory is applied by treating the asymptotic results as approximately valid for finite sample sizes as well. Such approach is often criticized for not having any mathematical grounds behind it, yet it is used ubiquitously anyway. The importance of the asymptotic theory is that it often makes possible to carry out the analysis and state many results which cannot be obtained within the standard '''“finite-sample theory”'''.
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==Overview==
Most statistical problems begin with a dataset of [[sample size|size]] ''n''. The asymptotic theory proceeds by assuming that it is possible to keep collecting additional data, so that the sample size would grow indefinitely:
: <math>
    n \to \infty\,
  </math>
Under this assumption many results can be obtained that are unavailable for samples of finite sizes. As an example consider the [[law of large numbers]]. This law states that for a sequence of [[iid]] random variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, , the sample averages <math style="vertical-align:-.1em">\scriptstyle\overline{X}_n</math> converge in probability to the population mean E[''X<sub>i</sub>''] as ''n'' → ∞. At the same time for finite ''n'' it is impossible to claim anything about the distribution of <math style="vertical-align:-.1em">\scriptstyle\overline{X}_n</math> if the distributions of individual ''X''<sub>''i''</sub>’s is unknown.
 
For various models slightly different modes of asymptotics may be used:
* For [[cross-sectional data]] ([[iid]]) the new observations are sampled independently, from the same fixed distribution. This is the standard case of {{nowrap|''n'' → ∞}} asymptotics.
* For [[longitudinal data]] ([[time series]]) the sampling method may differ from model to model. Sometimes the data is assumed to be [[ergodic process|ergodic]], in other applications it can be [[integrated process|integrated]] or [[cointegration|cointegrated]]. In this case the asymptotic is again taken as the number of observations (usually denoted ''T'' in this case) goes to infinity: {{nowrap|''T'' → ∞}}.
* For [[panel data]], it is commonly assumed that one dimension in the data (''T'') remains fixed, whereas the other dimension grows: {{nowrap|1=''T'' = const}}, {{nowrap|''n'' → ∞}}.
 
Besides these standard approaches, various other '''“alternative” asymptotic''' approaches exist:
* Within the [[local asymptotic normality]] framework, it is assumed that the value of the “true parameter” in the model varies slightly with ''n'', such that the ''n''-th model corresponds to <math style="vertical-align:-.3em">\scriptstyle\theta_n\,=\,\theta+h/\sqrt{n}</math>. This approach lets us study the [[regular estimator|regularity of estimators]].
* When [[statistical test]]s are studied for their power to distinguish against the alternatives that are close to the null hypothesis, it is done within the so-called ''“local alternatives”'' framework: the null hypothesis is ''H''<sub>0</sub>: ''θ'' = ''θ''<sub>0</sub>, and the alternative is ''H''<sub>1</sub>: <math style="vertical-align:-.3em">\scriptstyle\theta\,=\,\theta_0+h/\sqrt{n}</math>. This approach is especially popular for the [[unit root test]]s.
* There are models where the dimension of the parameter space Θ<sub>''n''</sub> slowly expands with ''n'', reflecting the fact that the more observations a statistician has, the more he is tempted to introduce additional parameters in the model. An example of this is the [[weak instrument]]s asymptotic.
* In [[kernel density estimation]] and [[kernel regression]] additional parameter — the bandwidth ''h'' — is assumed. In these models it is typically taken that ''h'' → 0 as ''n'' → ∞, however the rate of convergence must be chosen carefully, usually ''h'' ∝ ''n''<sup>−1/5</sup>.
 
==Modes of convergence of random variables==
{{see|Convergence of random variables}}
 
==Asymptotic properties==
===Estimators===
 
*  '''[[Consistent estimator|Consistency]]''': a sequence of estimators is said to be ''consistent'', if it [[converges in probability]] to the true value of the parameter being estimated:
: <math>
    \hat\theta_n\ \xrightarrow{p}\ \theta_0
  </math>
Generally an [[estimator]] is just some, more or less arbitrary, function of the data. The property of consistency requires that the estimator was estimating the quantity we intended it to. As such, it is the most important property in the estimation theory: estimators that are known to be inconsistent are never used in practice.
 
*  '''[[Asymptotic distribution]]''': if it is possible to find sequences of non-random constants {''a<sub>n</sub>''}, {''b<sub>n</sub>''} (possibly depending on the value of ''θ''<sub>0</sub>), and a non-degenerate distribution ''G'' such that
: <math>
    b_n(\hat\theta_n - a_n)\ \xrightarrow{d}\ G ,
  </math>
then the sequence of estimators <math style="vertical-align:-.1em">\scriptstyle\hat\theta_n</math> is said to have the ''asymptotic distribution'' ''G''.
 
Most often, the estimators encountered in practice have the '''asymptotically normal''' distribution, with {{nowrap|1=''a<sub>n</sub>'' = ''θ''<sub>0</sub>}}, {{nowrap|1=''b<sub>n</sub>'' = √''n''}}, and {{nowrap|1=''G'' = [[normal distribution|''N''(0, ''V'')]]}}:
: <math>
    \sqrt{n}(\hat\theta_n - \theta_0)\ \xrightarrow{d}\ \mathcal{N}(0, V).
  </math>
 
*  '''Asymptotic [[confidence region]]s'''.
*  '''[[Regular estimator|Regularity]]'''.
 
==Asymptotic theorems==
* '''[[Law of large numbers]]'''
* '''[[Central limit theorem]]'''
* '''[[Slutsky’s theorem]]'''
* '''[[Continuous mapping theorem]]'''
 
==Notes==
{{reflist}}
 
{{refimprove|date=April 2013}}
{{inline|date=April 2013}}
==References==
{{refbegin}}
* {{cite book
  | last1 = Le Cam | first1 = Lucien | authorlink1 = Lucien Le Cam
  | last2 = Yang | first2 = Grace Lo
  | title = Asymptotics in statistics (some basic concepts)
  | year = 2000
  | edition = 2nd
  | publisher = Springer
  | isbn = 0-387-95036-2
  | ref = CITEREFLe_CamYang2000
  }}
* {{cite book
  | last = van der Vaart | first = A.W.
  | title = Asymptotic statistics
  | year = 1998
  | publisher = Cambridge University Press
  | isbn = 978-0-521-49603-2
  | ref = CITEREFvan_der_Vaart1998
  }}
{{refend}}
 
{{DEFAULTSORT:Asymptotic Theory (Statistics)}}
[[Category:Statistical theory]]
[[Category:Econometrics]]
[[Category:Asymptotic statistical theory]]

Latest revision as of 20:25, 11 August 2014

There are also many businesses that are now offering the fixed gear bike, kromica fixie and takara fixie bikes, much to the delight of many biking aficionados. Above the very last ten many years the trails of this place have become smaller and scaled-down. Check out bicycle repair tool reviews of the most popular tools that provide the best value for your dollar. To check out more in regards to Size guaide mountain bike sizing. look into the web page. noted, Valhalla was designed "by the people from Whistler" (the company Gravity Logic) so "it's special. Why would I want to change, even for a day, the most important and shaping event in my life.

The prices offered by these online stores are very competitive and realistic according to the market. in summary let me give some procuations you bshould take when ridding it , ridding a bike at high speed can lead to accidents just maintain a minimum and also follow the traffic lights , if it is for sport just make enough practice and enjoy the game. The price you can expect to pay for stems is actually determined by the materials used along with its weight. - Get a bike that is lightweight and made out of carbon fiber materials. The micro-brewery and pub sits right along the White Pine Trail, a paved rail-trail that runs from Grand Rapids to Cadillac.

If you are looking for the best safety mountain bike accessories then the first thing you need to look into is a helmet. If you do it as soon as you get home, then its done and you can concentrate on eating and relaxing. There are great coaches, who will train you to tweet like a bird in spring or Link in, befriend and get a fan page. Mounting biking is a fun sport for professionals and beginners alike. The bike shop personnel should be able to advise you on how to choose the right frame size.

For more info about Anyclean and the range of cleaning services on offer please visit:. This article discusses how to go about buying a mountain bike. It is primarily design of Billet Motorola clutch kit that replaces your stock clutch within your GPX 150, 160 and additionally YX 140, 150, 160. The dedication and solid experience of the experts, it is not difficult to accomplish such goal. With Adventure Kerala ( you will enjoy adventure sports like mountaineering, rapelling, rock climbing, river crossing, and a whole lot of adventure activities.

In the recent article section, you can find the list of biking related articles with a detailed description present on the official website. It's usually best to buy from your bike store to begin with, as they can help you fit the bike and give you advice. Trying out various sizes is the first step in choosing the correct folding bike. Many of these trails and stunts are so dangerous that signs are up because of the potential of a serious injury or even death. The popular Whistler mountain bike website now has an extension into iphone.