Empirical Processes

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The empirical process refers to the idea to obtain nicely behaved functions of data. In a simple example the empirical process is defined as $\sqrt{n}\left(F_n(x) - F(x)\right)$, where $F_n(x)$ is the empirical distribution function and $F(x)$ is the distribution function. The empirical distribution function scaled by $n$ is $$nF_{n}(x)=\sum_{i=1}^n 1\{X_i\le x\} $$ has a binomial distribution with mean $nF(x)$. Then by the central limit theorem we have $$\sqrt{n}\left(F_n(x)-F(x)\right) \to N(0,F(x)(1-F(x))).$$
In empirical processes this type of convergence is made more general by considering random functions instead of .

References

• Sara van de Geer (2006). Empirical process theory and applications. (available online) 
• David Pollard (1990). Empirical processes: Theory and applications. IMS (available online)
• Aad van der Vaart (1998). Asymptotic Statistics. Cambridge University Press.