Questions tagged [empirical-processes]
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Conditional Density Estimation in RKHS
I would like to model the conditional density of two real-valued random variable and estimate it using the empirical conditional mean embedding. I am not sure which of these two are correct way of ...
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Is this class of indicator functions Donsker?
Consider the following class of indicator functions $\mathcal{F}=\{1_{f(x,\xi)\leq 0}: x\in \mathcal{X}\}$, where $\xi$ is a random vector, $\mathcal{X}\subset \mathbb{R}^d$ is a compact set. We may ...
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Generalization error bound for Empirical Risk minimizer on Gaussian noisy data
I have datapoints that are sampled from a distribution $\mathbb{D}$. Each datapoint is a tuple $(t,y)$ of a time $t \in [0,T]$ that is sampled uniformly and a value $y(t) \sim u(t) + \mathcal{N}(0, \...
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Convergence of integrals wrt empirical distribution
I have iid samples $X_1, \dots, X_n$ from some distribution $\mathcal{P}$. Based on them I can construct an empirical distribution $\mathcal{P}_n$, where
$$\mathcal{P}_n (A) = \frac{1}{n} \sum^n_{i = ...
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Uniform Convergence of Empirical Means: Universal Separability Implies Measurability
On page 38 of Pollard's book "Convergence of Stochastic Processes" one finds the following exercise (Problem 3 in Chapter II):
Call a class of functions $\mathcal{F}$ universally separable ...
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Confusion about LLN for Empirical Processes/Measures
I have seen the following LLN result on empirical measures:
(Statement 1) Let $X, X_1, X_2, \cdots, X_n$ be i.i.d. random variables taking values in $[0, 1]$. Then
$$
\mathbb{E}\sup_{f \in \mathcal{F}...
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Empirical process symmetrization lower bound
This is part of Exercise 7.1.9 from Vershynin's book "High Dimensional Probability":
Suppose $X_1(t),\dots,X_N(t)$ are $N$ independent, mean zero random processes indexed by points $t\in T$....
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Do we need to assume that $y$ is bounded
Suppose that $X_1,\dots, X_n$ are iid $P$ on $\mathcal{X}$. The empirical measure $\mathbb{P}_n$ is defined by $$\mathbb{P}_n:=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$
For a real-valued function $f$ on $\...
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How to prove that every Donsker class is also a Glivenko-Cantelli class
From https://en.wikipedia.org/wiki/Vapnik–Chervonenkis_theory
Let $X_1, ..., X_n$ be a random sample from a probability distribution P. Let $$P_nf=\frac{1}{n}\sum_{i=1}^nf(X_i), \quad Pf=\int fdP.$$ A ...
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Prove Glivenko-Cantelli (GC) by bracketing
Question: My question about the theorem Glivenko-Cantelli (GC) by bracketing is: if $N_{[]}(\varepsilon,\mathcal{F},L_{1}(P))=\infty$,
which part of the above proof will fail, thus GC will fail.
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The asymptotic distribution of the criterion function in M-estimation
Suppose that we are interested in a parameter (or functional)
$\theta$ attached to the distribution of observations $\left\{ X_{1},\ldots,X_{n}\right\} $.
A popular method for finding an estimator $\...
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Bracketing entropy for unimodal functions
This is exercise 2.3 from van de Geer's Empirical processes in M-estimation.
Let $\mathcal G$ be the class of unimodal functions $g : \mathbb R \rightarrow [0, 1]$. A function $f$ is unimodal if there ...
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Joint measurability of discrete integral.
I'm currently considering the following map: Let $\mathcal{F}$ be a subset of some $L^2$ space with respect to a probability measure on $\mathbb{R}$ and let $X=(X_1,X_2,\ldots,X_n)$ be points on the ...
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Translation of box type ball to center it in a convex set but maintaining the covering properties
I have the following problem. Suppose to have a cube $Q$ and a symmetric box type ball $B$ whose center is outside Q, but such that $Q\cap B \ne\emptyset$. Furthermore $B$ is not oriented like $Q$. I ...
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Does the empirical c.d.f. converge to the population c.d.f. when the data is drawn from a mixing stochastic process
For i.i.d. observations, the Dvoretzky–Kiefer–Wolfowitz inequality tells us that the empirical c.d.f. almost surely converges to the population c.d.f. (i.e., $\underset{x \in \mathbb{R}}{\sup} | F_n(x)...
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