Related papers: Local Dvoretzky-Kiefer-Wolfowitz confidence bands
Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set…
Gowers introduced, for d\geq 1, the notion of dimension-d uniformity U^d(f) of a function f: G -> \C, where G is a finite abelian group and \C are the complex numbers. Roughly speaking, if U^d(f) is small, then f has certain…
We analyze the (unconditional) distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. First, we derive the exact finite-sample cumulative distribution function (cdf) of…
A sum of lognormal random variables (RVs) appears in many problems of science and engineering. For example, it is invloved in computing the distribution of recevied signal and interference powers for radio channels subject to lognormal…
We introduce a maximal inequality for a local empirical process under strongly mixing data. Local empirical processes are defined as the (local) averages $\frac{1}{nh}\sum_{i=1}^n \mathbf{1}\{x - h \leq X_i \leq x+h\}f(Z_i)$, where $f$…
Under the assumption that the distribution of a nonnegative random variable $X$ admits a bounded coupling with its size biased version, we prove simple and strong concentration bounds. In particular the upper tail probability is shown to…
The Fast Diffusion Equation (FDE) $u_t= \Delta u^m$, with $m\in (0,1)$, is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy-Dirichlet problem posed on smooth bounded Euclidean…
A classical result in the theory of Loewner's parametric representation states that the semigroup $\mathfrak U_*$ of all conformal self-maps $\phi$ of the unit disk $\mathbb{D}$ normalized by $\phi(0) = 0$ and $\phi'(0) > 0$ can be obtained…
In this paper, we investigate the cumulative distribution functions (CDFs) of the maximum and minimum of multivariate Poisson distributions with three dependence structures, namely, the common shock, comonotonic shock and…
Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is…
We introduce a generalized version of the local Lipschitz number $\textrm{lip}\,u$, and show that it can be used to characterize Sobolev functions $u\in W_{\textrm{loc}}^{1,p}(\mathbb R^n)$, $1\le p\le \infty$, as well as functions of…
Let $\mu$ be a probability measure on $\mathbb{R}$ with cumulative distribution function $F$, $(x_{i})_{1}^{n}$ a large i.i.d. sample from $\mu$, and $F_{n}$ the associated empirical distribution function. The Glivenko-Cantelli theorem…
We derive concentration inequalities for the supremum norm of the difference between a kernel density estimator (KDE) and its point-wise expectation that hold uniformly over the selection of the bandwidth and under weaker conditions on the…
We leverage neural networks as universal approximators of monotonic functions to build a parameterization of conditional cumulative distribution functions (CDFs). By the application of automatic differentiation with respect to response…
This paper presents the probability distribution function (p.d.f.) and cumulative distribution function (c.d.f.) of the weighted sum of central independent chi-squared random variables with non-zero weighs based on a method using moment…
In the context of treatment effect estimation, this paper proposes a new methodology to recover the counterfactual distribution when there is a single (or a few) treated unit and possibly a high-dimensional number of potential controls…
In this paper, we mainly discuss the local regularity of the solution to the following problem \begin{align*} \begin{cases} -\dive({\bf{A}}(x)\nabla u(x))=f(x),&~x\in\Omega,\\ u(x)=0,&~x\in\partial\Omega, \end{cases} \end{align*} where…
Statistical model checking (SMC) randomly samples probabilistic models to approximate quantities of interest with statistical error guarantees. It is traditionally used to estimate probabilities and expected rewards, i.e. means of different…
While it is a common knowledge that AC coefficients of Fourier-related transforms, like DCT-II of JPEG image compression, are from Laplace distribution, there was tested more general EPD (exponential power distribution) $\rho\sim…
Let $n\geq 1$, $K>0$, and let $X=(X_1,X_2,\dots,X_n)$ be a random vector in $\mathbb{R}^n$ with independent $K$--subgaussian components. We show that for every $1$--Lipschitz convex function $f$ in $\mathbb{R}^n$ (the Lipschitzness with…