Compact hypothesis and extremal set estimators

João Tiago Mexia; Pedro Corte Real

Displaying similar documents to “Compact hypothesis and extremal set estimators”

Existence, Consistency and computer simulation for selected variants of minimum distance estimators

Václav Kůs, Domingo Morales, Jitka Hrabáková, Iva Frýdlová (2018)

Kybernetika

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The paper deals with sufficient conditions for the existence of general approximate minimum distance estimator (AMDE) of a probability density function $f_{0}$ on the real line. It shows that the AMDE always exists when the bounded $φ$ -divergence, Kolmogorov, Lévy, Cramér, or discrepancy distance is used. Consequently, $n^{- 1 / 2}$ consistency rate in any bounded $φ$ -divergence is established for Kolmogorov, Lévy, and discrepancy estimators under the condition that the degree of variations of the corresponding...

Estimator selection in the gaussian setting

Yannick Baraud, Christophe Giraud, Sylvie Huet (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$ with independent components of common unknown variance $σ^{2}$ . Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection $𝔽$ of estimators of $f$ based on $Y$ and, with the same data $Y$ , aim at selecting an estimator among $𝔽$ with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to $Y$ may be unknown....

Orthogonal series regression estimation under long-range dependent errors

Waldemar Popiński (2001)

Applicationes Mathematicae

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This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample $Y_{i} = f (X_{i}) + η_{i}$ , i=1,...,n, where $X_{i} \in A \subset ℝ^{d}$ are independently chosen from a distribution with density ϱ ∈ L¹(A) and $η_{i}$ are zero mean stationary errors with long-range dependence. Convergence rates of the error $n^{- 1} \sum_{i = 1}^{n} (f (X_{i}) - f ̂_{N} (X_{i})) ²$ for the estimator $f ̂_{N} (x) = \sum_{k = 1}^{N} c ̂_{k} e_{k} (x)$ , constructed using an orthonormal system...

Optimal estimators in learning theory

V. N. Temlyakov (2006)

Banach Center Publications

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This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs $x_{i}$ and outputs $y_{i}$ , i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator $f_{z}$ on the base of given data $z : = ((x ₁, y ₁), . . ., (x_{m}, y_{m}))$ that approximates well the regression function...

Spatially adaptive density estimation by localised Haar projections

Florian Gach, Richard Nickl, Vladimir Spokoiny (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Given a random sample from some unknown density $f_{0} : ℝ \to [0, \infty)$ we devise Haar wavelet estimators for $f_{0}$ with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny ( (1997) 927–947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of $f_{0}$ , simultaneously for every point $x$ in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in...

On orthogonal series estimation of bounded regression functions

Waldemar Popiński (2001)

Applicationes Mathematicae

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The problem of nonparametric estimation of a bounded regression function $f \in L ² ({[a, b]}^{d})$ , [a,b] ⊂ ℝ, d ≥ 1, using an orthonormal system of functions $e_{k}$ , k=1,2,..., is considered in the case when the observations follow the model $Y_{i} = f (X_{i}) + η_{i}$ , i=1,...,n, where $X_{i}$ and $η_{i}$ are i.i.d. copies of independent random variables X and η, respectively, the distribution of X has density ϱ, and η has mean zero and finite variance. The estimators are constructed by proper truncation of the function $f ̂ ₙ (x) = \sum_{k = 1}^{N (n)} c ̂_{k} e_{k} (x)$ , where the coefficients $c ̂ ₁, . . ., c ̂_{N (n)}$ ...

Universal rates for estimating the residual waiting time in an intermittent way

Gusztáv Morvai, Benjamin Weiss (2020)

Kybernetika

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A simple renewal process is a stochastic process ${X_{n}}$ taking values in ${0, 1}$ where the lengths of the runs of $1$ ’s between successive zeros are independent and identically distributed. After observing $X_{0}, X_{1}, ... X_{n}$ one would like to estimate the time remaining until the next occurrence of a zero, and the problem of universal estimators is to do so without prior knowledge of the distribution of the process. We give some universal estimates with rates for the expected time to renewal as well as for the conditional...

On the strong Brillinger-mixing property of $α$ -determinantal point processes and some applications

Lothar Heinrich (2016)

Applications of Mathematics

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First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C (x, y)$ defining an $α$ -determinantal point process (DPP). Assuming absolute integrability of the function $C_{0} (x) = C (o, x)$ , we show that a stationary $α$ -DPP with kernel function $C_{0} (x)$ is “strongly” Brillinger-mixing, implying, among others, that its tail- $σ$ -field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch...

Density estimation via best $L^{2}$ -approximation on classes of step functions

Dietmar Ferger, John Venz (2017)

Kybernetika

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We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best $L^{2}$ -approximation of a probability density function $f$ . If $f$ itself is a step-function the number of jumps may be unknown.

Robin functions and extremal functions

T. Bloom, N. Levenberg, S. Ma'u (2003)

Annales Polonici Mathematici

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Given a compact set $K \subset ℂ^{N}$ , for each positive integer n, let $V^{(n)} (z)$ = $V_{K}^{(n)} (z)$ := sup $1 / (d e g p) V_{p (K)} (p (z))$ : p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V_{K}^{(n)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_{K} (z)$ := max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${| | p | |}_{K} \leq 1$ ]. Our main result is that if K is regular, then all of the functions $V_{K}^{(n)}$ are continuous; and their associated Robin functions $ϱ_{V_{K}^{(n)}} (z) : = l i m s u p_{| λ | \to \infty} [V_{K}^{(n)} (λ z) - l o g (| λ |)]$ increase to $ϱ_{K} : = ϱ_{V_{K}}$ for all z outside a pluripolar...

Estimating composite functions by model selection

Yannick Baraud, Lucien Birgé (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the problem of estimating a function $s$ on ${[- 1, 1]}^{k}$ for large values of $k$ by looking for some best approximation of $s$ by composite functions of the form $g \circ u$ . Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions $g, u$ and statistical frameworks. In particular, we handle the problems of approximating $s$ by additive functions, single and multiple index models, artificial neural networks, mixtures...

Notes on strongly Whyburn spaces

Masami Sakai (2016)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the notion of a strongly Whyburn space, and show that a space $X$ is strongly Whyburn if and only if $X \times (ω + 1)$ is Whyburn. We also show that if $X \times Y$ is Whyburn for any Whyburn space $Y$ , then $X$ is discrete.

More on exposed points and extremal points of convex sets in $ℝ^{n}$ and Hilbert space

Stoyu T. Barov (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let $𝕍$ be a separable real Hilbert space, $k \in ℕ$ with $k < dim 𝕍$ , and let $B$ be convex and closed in $𝕍$ . Let $𝒫$ be a collection of linear $k$ -subspaces of $𝕍$ . A point $w \in B$ is called exposed by $𝒫$ if there is a $P \in 𝒫$ so that $(w + P) \cap B = {w}$ . We show that, under some natural conditions, $B$ can be reconstituted as the convex hull of the closure of all its exposed by $𝒫$ points whenever $𝒫$ is dense and $G_{δ}$ . In addition, we discuss the question when the set of exposed by some $𝒫$ points forms a $G_{δ}$ -set.