Estimation of a regression function on a Poisson process
type: Item
instance of: scholarly article
title: Estimation of a regression function on a Poisson process
zbMATH Open document ID: 0644.62045
DOI: 10.1016/0167-7152(87)90058-7
author: Galaye Dia
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
review text: Let \(f_ 0\) be an \(R_+\times R\)-valued point process, let \(f_ 0(A)\) be the total number of points of \(f_ 0\) falling in A, and let \(\ell =f_ 0(R_+\times R)\) be a.s. finite. The first projection of \(f_ 0\) is assumed to be a Poisson process. For \(\ell \geq 1\), let \((X_ 1,Y_ 1),(X_ 2,Y_ 2), ... ,(X_{\ell},Y_{\ell})\) be the points of the process \(f_ 0\). It is assumed that \[ \psi (x):=E(Y_ j| \quad X_ 1=x_ 1,...,X_{j-1}=x_{j-1},X_ j=x,X_{j+1}=x_{j+1},...,X_{\ell}=x\quad_{\ell}) \] depends neither on \(\ell\) nor on j. On the basis of n i.i.d. copies of \(f_ 0\), a simple estimator \(\psi_ n(x)\) of the regression function \(\psi\) (x) is constructed, and it is shown that \((\forall x\in R_+)\) \(E(\psi_ n(x)-\psi (x))^ 2\to 0\) as \(n \to\infty.\)
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62J02
Mathematics Subject Classification ID: 62M09
zbMATH DE Number: 4050769
zbMATH Keywords: sufficient conditions of weak consistency
zbMATH Keywords: fixed-bandwidth regressogram estimator
zbMATH Keywords: superposition
zbMATH Keywords: point process
zbMATH Keywords: Poisson process
reviewed by: Ryszard Zieliński
MaRDI profile type: Publication
full work available at URL: 0167-7152(87)90058-7
OpenAlex ID: W1993880757
cites work: Q4772487
cites work: Q3878552
cites work: Q4076615
cites work: Q3290184
description: scientific article; zbMATH DE number 4050769