Misplaced Pages

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) The topic of this article may not meet Misplaced Pages's general notability guideline. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted. Find sources: "Donsker classes" – news · newspapers · books · scholar · JSTOR (January 2024) (Learn how and when to remove this message) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Donsker classes" – news · newspapers · books · scholar · JSTOR (January 2024) (Learn how and when to remove this message) (Learn how and when to remove this message)

A class of functions is considered a Donsker class if it satisfies Donsker's theorem, a functional generalization of the central limit theorem.

Definition

Let ${\displaystyle {\mathcal {F}}}$ be a collection of square integrable functions on a probability space ${\displaystyle ({\mathcal {X}},{\mathcal {A}},P)}$. The empirical process ${\displaystyle \mathbb {G} _{n}}$ is the stochastic process on the set ${\displaystyle {\mathcal {F}}}$ defined by $${\displaystyle \mathbb {G} _{n}(f)={\sqrt {n}}(\mathbb {P} _{n}-P)(f)}$$ where ${\displaystyle \mathbb {P} _{n}}$ is the empirical measure based on an iid sample ${\displaystyle X_{1},\dots ,X_{n}}$ from ${\displaystyle P}$.

The class of measurable functions ${\displaystyle {\mathcal {F}}}$ is called a Donsker class if the empirical process ${\displaystyle (\mathbb {G} _{n})_{n=1}^{\infty }}$ converges in distribution to a tight Borel measurable element in the space ${\displaystyle \ell ^{\infty }({\mathcal {F}})}$.

By the central limit theorem, for every finite set of functions ${\displaystyle f_{1},f_{2},\dots ,f_{k}\in {\mathcal {F}}}$, the random vector ${\displaystyle (\mathbb {G} _{n}(f_{1}),\mathbb {G} _{n}(f_{2}),\dots ,\mathbb {G} _{n}(f_{k}))}$ converges in distribution to a multivariate normal vector as ${\displaystyle n\rightarrow \infty }$. Thus the class ${\displaystyle {\mathcal {F}}}$ is Donsker if and only if the sequence ${\displaystyle (\mathbb {G} _{n})_{n=1}^{\infty }}$ is asymptotically tight in ${\displaystyle \ell ^{\infty }({\mathcal {F}})}$

Examples and Sufficient Conditions

Classes of functions which have finite Dudley's entropy integral are Donsker classes. This includes empirical distribution functions formed from the class of functions defined by ${\displaystyle \mathbb {I} _{(-\infty ,t]}}$ as well as parametric classes over bounded parameter spaces. More generally any VC class is also Donsker class.

Properties

Classes of functions formed by taking infima or suprema of functions in a Donsker class also form a Donsker class.

Donsker's Theorem

Donsker's theorem states that the empirical distribution function, when properly normalized, converges weakly to a Brownian bridge—a continuous Gaussian process. This is significant as it assures that results analogous to the central limit theorem hold for empirical processes, thereby enabling asymptotic inference for a wide range of statistical applications.

The concept of the Donsker class is influential in the field of asymptotic statistics. Knowing whether a function class is a Donsker class helps in understanding the limiting distribution of empirical processes, which in turn facilitates the construction of confidence bands for function estimators and hypothesis testing.

See also

• Empirical process
• Central limit theorem
• Brownian bridge
• Glivenko–Cantelli theorem
• Vapnik–Chervonenkis theory
• Weak convergence (probability)

References

1. van der Vaart, A. W.; Wellner, Jon A. (2023). Weak Convergence and Empirical Processes. Springer Series in Statistics. p. 139. doi:10.1007/978-3-031-29040-4. ISBN 978-3-031-29038-1.
2. ^ Vaart AW van der. Asymptotic Statistics. Cambridge University Press; 1998.
3. ^ van der Vaart, A. W., & Wellner, J. A. (1996). Weak Convergence and Empirical Processes. In Springer Series in Statistics. Springer New York. https://doi.org/10.1007/978-1-4757-2545-2

• Probability theory
• Central limit theorem