Polygonal smoothing of the empirical distribution function
type: Item
instance of: scholarly article
title: Polygonal smoothing of the empirical distribution function
zbMATH Open document ID: 1401.62050
author: Delphine Blanke
author: Denis Bosq
published in: Statistical Inference for Stochastic Processes
publication date: 2018-08-10T00:00:00Z
full work available at URL: polygonal cdf preprint.pdf
review text: Let \( X_{1},\ldots,X_{n} \) be i.i.d. random variables with cumulative distribution function (cdf) \(F\). The most classical and simple estimate of the cdf \(F\) is the empirical distribution function (edf) \( F_{n}\left(x \right) = \frac{1}{n} \sum_{i=1}^{n} \mathbb{I}_{\left( -\infty,x\right] } \left(X_{i} \right) \), \( x\in \mathbb{R} \), where \( \mathbb{I}_{A} \) denotes the indicator function of the set \(A\). In the case, when \(F\) is absolutely continuous, the edf \(F_{n}\) is not continuous. There are several smoothing techniques to correct this drawback. For estimating \(F\), the authors consider a polygonal smoothing of the edf \(F_{n}\). From the authors' abstract: We present two families of polygonal estimators of the distribution function: the first family is based on the knowledge of the support while the second addresses the case of an unknown support. Polygonal smoothing is a simple and natural method for regularizing the empirical distribution function \(F_n\) but its properties have not been studied deeply. First, consistency and exponential type inequalities are derived from well-known convergence properties of \(F_n\). Then, we study their mean integrated squared error (MISE) and we establish that polygonal estimators may improve the MISE of \(F_n\). We conclude by some numerical results to compare these estimators globally, and also together with the integrated kernel distribution estimator.
reviewed by: Wiesław Dziubdziela
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 62G20
zbMATH DE Number: 6916511
zbMATH Keywords: distribution function estimation
zbMATH Keywords: polygonal estimator
zbMATH Keywords: cumulative frequency polygon
zbMATH Keywords: order statistics
zbMATH Keywords: mean integrated squared error
zbMATH Keywords: exponential inequality
zbMATH Keywords: smoothed processes
describes a project that uses: kerdiest
describes a project that uses: R
MaRDI profile type: Publication
OpenAlex ID: W2800722727
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Wikidata QID: Q129817698
DOI: 10.1007/S11203-018-9183-Y
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description: scientific article; zbMATH DE number 6916511