stochastic equicontinuity

{{Expand section|date=September 2010|{{JSTOR|2938179}}}}

Let $\backslash \{\; H\_n(\backslash theta):\; n\; \backslash geq\; 1\; \backslash \}$ be a family of random functions defined from $\backslash Theta\; \backslash rightarrow\; \backslash reals$, where $\backslash Theta$ is any normed metric space. Here $\backslash \{\; H\_n(\backslash theta)\; \backslash \}$ 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 $\backslash \{\; H\_n(\backslash theta)\; \backslash \}$, 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 $\backslash \{\; H\_n\; \backslash \}$ is stochastically equicontinuous if, for every $\backslash epsilon\; >\; 0$ and $\backslash eta\; >\; 0$, there is a $\backslash delta\; >\; 0$ such that:

:

Here B(θ, δ) represents a ball in the parameter space, centred at θ and whose radius depends on δ.

• M-Estimators: Stochastic equicontinuity is needed to prove the consistency and asymptotic normality of M-estimators.{{Cite web |date=15 February 2007 |title=Applications of ULLNs: Consistency of M-estimators |url=https://people.eecs.berkeley.edu/~jordan/courses/210B-spring08/lectures/stat210b_lecture_9.pdf}}

Example: Consider an M-estimator defined by minimizing a sample objective function $Q\_n(\backslash theta)$. Stochastic equicontinuity helps in showing that $Q\_n(\backslash theta)$ converges uniformly to its population counterpart $Q(\backslash theta)$, ensuring that the estimator $\backslash hat\{\backslash theta\}\_n$ converges to the true parameter $\backslash theta\_0$.

• Nonparametric Estimation: In nonparametric estimation, stochastic equicontinuity is needed in establishing the uniform convergence of nonparametric estimators. Like - kernel density estimators or spline regressions.{{Cite web |date=30 August 2010 |title=Uniform Convergence in Probability and Stochastic Equicontinuity |url=https://users.ssc.wisc.edu/~xshi/Newey1991.pdf}}

Example: For a kernel density estimator $\backslash hat\{f\}\_n(x)$, stochastic equicontinuity ensures that $\backslash hat\{f\}\_n(x)$ converges uniformly to the true density function $f(x)$ as the sample size $n$ increases.

• Nonlinear Time Series Models: In nonlinear time series models, stochastic equicontinuity ensures the stability and consistency of estimators.

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.{{Cite journal |last=Hagemann |first=Andreas |title=Stochastic equicontinuity in nonlinear time series models |journal=The Econometrics Journal |date=2014 |volume=17 |pages=188–196 |url=https://academic.oup.com/ectj/article-abstract/17/1/188/5056416 |language=English |doi=10.1111/ectj.12013|arxiv=1206.2385 }}

• Consistency of Estimators: Stochastic equicontinuity is a key condition for proving the consistency of estimators in time series models.

{{Reflist}}

• {{cite book |first=David |last=Pollard |title=Convergence of Stochastic Processes |location=New York |publisher=Springer |year=1984 |chapter=Stochastic Equicontinuity |pages=138–142 |isbn=0-387-90990-7 |chapter-url=https://books.google.com/books?id=B2vgGMa9vd4C&pg=PA138 }}

Category:Asymptotic theory (statistics)

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