statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

, asymptotic theory, or large sample theory, is a framework for assessing properties of

estimator In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...

s and

statistical test A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...

s. Within this framework, it is often assumed that the

sample size Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populati ...

may grow indefinitely; the properties of estimators and tests are then evaluated under the limit of . In practice, a limit evaluation is considered to be approximately valid for large finite sample sizes too.Höpfner, R. (2014), Asymptotic Statistics, Walter de Gruyter. 286 pag. ,

Overview

Most statistical problems begin with a dataset of

size Size in general is the magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions ( length, width, height, diameter, perimeter), area, or volume. Size can also be me ...

. The asymptotic theory proceeds by assuming that it is possible (in principle) to keep collecting additional data, thus that the sample size grows infinitely, i.e. . Under the assumption, many results can be obtained that are unavailable for samples of finite size. An example is the

weak law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...

. The law states that for a sequence of

independent and identically distributed In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...

(IID)

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

s , if one value is drawn from each random variable and the average of the first values is computed as , then the converge in probability to the population mean as .A. DasGupta (2008), ''Asymptotic Theory of Statistics and Probability'', Springer. , In asymptotic theory, the standard approach is . For some

statistical model A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...

s, slightly different approaches of asymptotics may be used. For example, with

panel data In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time. Time series and ...

, it is commonly assumed that one dimension in the data remains fixed, whereas the other dimension grows: and , or vice versa. Besides the standard approach to asymptotics, other alternative approaches exist: * Within the

local asymptotic normality In statistics, local asymptotic normality 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 th ...

framework, it is assumed that the value of the "true parameter" in the model varies slightly with , such that the -th model corresponds to . This approach lets us study the regularity of estimators. * When

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 and the alternative is . This approach is especially popular for the

unit root test In statistics, a unit root test tests whether a time series variable is non-stationary and possesses a unit root. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either stationarity, trend ...

s. * There are models where the dimension of the parameter space slowly expands with , reflecting the fact that the more observations there are, the more structural effects can be feasibly incorporated in the model. * In

kernel density estimation In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on '' kernels'' as ...

and

kernel regression In statistics, kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable. The objective is to find a non-linear relation between a pair of random variables ''X'' and ''Y''. In any nonparametric ...

, an additional parameter is assumed—the bandwidth . In those models, it is typically taken that as . The rate of convergence must be chosen carefully, though, usually . In many cases, highly accurate results for finite samples can be obtained via numerical methods (i.e. computers); even in such cases, though, asymptotic analysis can be useful. This point was made by , as follows.

Modes of convergence of random variables

Asymptotic properties

Estimators

''
Consistency In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
''

A sequence of estimates is said to be ''consistent'', if it converges in probability to the true value of the parameter being estimated: :

\hat\theta_n\ \xrightarrow\ \theta_0.

That is, roughly speaking with an infinite amount of data the

(the formula for generating the estimates) would almost surely give the correct result for the parameter being estimated.

''
Asymptotic distribution In mathematics and statistics, an asymptotic distribution is a probability 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 ...
''

If it is possible to find sequences of non-random constants , (possibly depending on the value of ), and a non-degenerate distribution such that :

b_n(\hat\theta_n - a_n)\ \xrightarrow\ G ,

then the sequence of estimators

\textstyle\hat\theta_n

is said to have the ''

asymptotic distribution In mathematics and statistics, an asymptotic distribution is a probability 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 ...

'' ''G''. Most often, the estimators encountered in practice are asymptotically normal, meaning their asymptotic distribution is the

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

, with , , and : :

\sqrt(\hat\theta_n - \theta_0)\ \xrightarrow\ \mathcal(0, V).

''Asymptotic
confidence region In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. It is a set of points in an ''n''-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, a ...
s''

Asymptotic theorems

Central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...

Continuous mapping theorem In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps converg ...

Glivenko–Cantelli theorem In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empir ...

Law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...

Law of the iterated logarithm In probability theory, the law of the iterated logarithm describes the magnitude of the fluctuations of a random walk. The original statement of the law of the iterated logarithm is due to A. Ya. Khinchin (1924). Another statement was given by ...

Slutsky's theorem In probability theory, Slutsky’s theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. The theorem was named after Eugen Slutsky. Slutsky's theorem is also attributed t ...

Delta method In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator. History The delta meth ...

References

Bibliography

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