The Bayes choice of an experiment in estimating a success probability

Alicja Jokiel-Rokita; Ryszard Magiera

Displaying similar documents to “The Bayes choice of an experiment in estimating a success probability”

Optimal estimators in learning theory

V. N. Temlyakov (2006)

Banach Center Publications

Similarity:

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...

Estimator selection in the gaussian setting

Yannick Baraud, Christophe Giraud, Sylvie Huet (2014)

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

Similarity:

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....

Estimation of the density of a determinantal process

Yannick Baraud (2013)

Confluentes Mathematici

Similarity:

We consider the problem of estimating the density $Π$ of a determinantal process $N$ from the observation of $n$ independent copies of it. We use an aggregation procedure based on robust testing to build our estimator. We establish non-asymptotic risk bounds with respect to the Hellinger loss and deduce, when $n$ goes to infinity, uniform rates of convergence over classes of densities $Π$ of interest.

A depth-based modification of the k-nearest neighbour method

Ondřej Vencálek, Daniel Hlubinka (2021)

Kybernetika

Similarity:

We propose a new nonparametric procedure to solve the problem of classifying objects represented by $d$ -dimensional vectors into $K \geq 2$ groups. The newly proposed classifier was inspired by the $k$ nearest neighbour (kNN) method. It is based on the idea of a depth-based distributional neighbourhood and is called $k$ nearest depth neighbours (kNDN) classifier. The kNDN classifier has several desirable properties: in contrast to the classical kNN, it can utilize global properties of the considered...

On orthogonal series estimation of bounded regression functions

Waldemar Popiński (2001)

Applicationes Mathematicae

Similarity:

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 $c ̂ ₁, . . ., c ̂_{N (n)}$ ...

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

Dietmar Ferger, John Venz (2017)

Kybernetika

Similarity:

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.

Minimax nonparametric prediction

Maciej Wilczyński (2001)

Applicationes Mathematicae

Similarity:

Let U₀ be a random vector taking its values in a measurable space and having an unknown distribution P and let U₁,...,Uₙ and $V ₁, . . ., V_{m}$ be independent, simple random samples from P of size n and m, respectively. Further, let $z ₁, . . ., z_{k}$ be real-valued functions defined on the same space. Assuming that only the first sample is observed, we find a minimax predictor d⁰(n,U₁,...,Uₙ) of the vector $Y^{m} = \sum_{j = 1}^{m} {(z ₁ (V_{j}), . . ., z_{k} (V_{j}))}^{T}$ with respect to a quadratic errors loss function.

Estimating composite functions by model selection

Yannick Baraud, Lucien Birgé (2014)

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

Similarity:

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...

Properties of unique information

Johannes Rauh, Maik Schünemann, Jürgen Jost (2021)

Kybernetika

Similarity:

We study the unique information function $U I (T : X ∖ Y)$ defined by Bertschinger et al. within the framework of information decompositions. In particular, we study uniqueness and support of the solutions to the convex optimization problem underlying the definition of $U I$ . We identify sufficient conditions for non-uniqueness of solutions with full support in terms of conditional independence constraints and in terms of the cardinalities of $T$ , $X$ and $Y$ . Our results are based on a reformulation of the first...

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

Gusztáv Morvai, Benjamin Weiss (2020)

Kybernetika

Similarity:

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...

Orthogonal series regression estimation under long-range dependent errors

Waldemar Popiński (2001)

Applicationes Mathematicae

Similarity:

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...

Evaluating default priors with a generalization of Eaton’s Markov chain

Brian P. Shea, Galin L. Jones (2014)

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

Similarity:

We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $𝛷$ be a class of functions on the parameter space and consider estimating elements of $𝛷$ under quadratic loss. If the formal Bayes estimator of every function in $𝛷$ is admissible, then the prior is strongly admissible with respect to $𝛷$ . Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with...

Orthogonal series estimation of band-limited regression functions

Waldemar Popiński (2014)

Applicationes Mathematicae

Similarity:

The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions $s_{k}$ , k = 0,±1,..., for the observation model $y_{j} = f (u_{j}) + η_{j}$ , j = 1,...,n, is considered, where f ∈ L²(ℝ) ∩ BL(Ω) for Ω > 0 is a band-limited function, $u_{j}$ are independent random variables uniformly distributed in the observation interval [-T,T], $η_{j}$ are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions...

A density version of the Carlson–Simpson theorem

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros (2014)

Journal of the European Mathematical Society

Similarity:

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k \geq 2$ and every set $A$ of words over $k$ satisfying $\lim \sup_{n \to \infty} | A \cap {[k]}^{n} | / k^{n} > 0$ there exist a word $c$ over $k$ and a sequence $(w_{n})$ of left variable words over $k$ such that the set $c \cup {c^{} w_{0} {(a_{0})}^{} . . .^{} w_{n} (a_{n}) : n \in ℕ and a_{0}, . . ., a_{n} \in [k]}$ is contained in $A$ . While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

Simplices rarely contain their circumcenter in high dimensions

Jon Eivind Vatne (2017)

Applications of Mathematics

Similarity:

Acute triangles are defined by having all angles less than $π / 2$ , and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension $n \geq 3$ , acuteness is defined by demanding that all dihedral angles between $(n - 1)$ -dimensional faces are smaller than $π / 2$ . However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property...

Theoretical analysis for $ℓ_{1}$ - $ℓ_{2}$ minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

Similarity:

We investigate the recovery of $k$ -sparse signals using the $ℓ_{1}$ - $ℓ_{2}$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$ -sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...