In
estimation theory
Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their valu ...
in
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, stochastic equicontinuity is a property 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 (estimation procedures) that is useful in dealing with their
asymptotic behaviour as the amount of data increases. It is a version of
equicontinuity
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
used in the context of functions of
random variables
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 ...
: that is,
random function
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...
s. The property relates to the rate of
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four Wei ...
of sequences of random variables and requires that this rate is essentially the same within a region of the
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for the ...
being considered.
For instance, stochastic equicontinuity, along with other conditions, can be used to show uniform weak convergence, which can be used to prove the
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four Wei ...
of
extremum estimator In statistics and econometrics, extremum estimators are a wide class of estimators for parametric models that are calculated through maximization (or minimization) of a certain objective function, which depends on the data. The general theory of ext ...
s.
Definition
Let
be a family of random functions defined from
, where
is any normed metric space. Here
might represent a sequence of estimators applied to datasets of size ''n'', given that the data arises from a population for which the parameter indexing the statistical model for the data is ''θ''. The randomness of the functions arises from the
data generating process under which a set of observed data is considered to be a realisation of a probabilistic or statistical model. However, in
, ''θ'' relates to the model currently being postulated or fitted rather than to an underlying model which is supposed to represent the mechanism generating the data. Then
is stochastically equicontinuous if, for every
and
, there is a
such that:
:
Here ''B''(''θ, δ'') represents a ball in the parameter space, centred at ''θ'' and whose radius depends on ''δ''.
References
Further reading
*
Asymptotic theory (statistics)
{{probability-stub