In estimation theory in statistics, stochastic equicontinuity is a property of estimators (estimation procedures) that is useful in dealing with their asymptotic behaviour as the amount of data increases. It is a version of equicontinuity used in the context of functions of random variables: that is, random functions. The property relates to the rate of convergence of sequences of random variables and requires that this rate is essentially the same within a region of the parameter space 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 of extremum estimators.
Definition.
Let formula_1 be a family of random functions defined from formula_2, where formula_3 is any normed metric space. Here formula_4 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 formula_4, "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 formula_6 is stochastically equicontinuous if, for every formula_7 and formula_8, there is a formula_9 such that:
formula_10
Here "B"("θ, δ") represents a ball in the parameter space, centred at "θ" and whose radius depends on "δ".
Applications.
Econometrics.
 Example: Consider an M-estimator defined by minimizing a sample objective function formula_11. Stochastic equicontinuity helps in showing that formula_11 converges uniformly to its population counterpart formula_13, ensuring that the estimator formula_14 converges to the true parameter formula_15.
 Example: For a kernel density estimator formula_16, stochastic equicontinuity ensures that formula_16 converges uniformly to the true density function formula_18 as the sample size formula_19 increases.
Time Series Models.
 Example: Consider a GARCH model used to model volatility in financial time series. Stochastic equicontinuity helps the estimated parameters of the GARCH model converge to the true parameters as the sample size increases, despite the model’s nonlinear nature.
References.
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