Coverage of credible intervals in nonparametric monotone regression
type: Item
instance of: scholarly article
title: Coverage of credible intervals in nonparametric monotone regression
zbMATH Open document ID: 1469.62232
DOI: 10.1214/20-AOS1989
author: Moumita Chakraborty
author: Subhashis Ghosal
published in: The Annals of Statistics
publication date: 2021-07-05T00:00:00Z
review text: This paper concerns a nonparametric regression model for a response variable \(Y\) with respect to a predictor variable \(X\in [0, 1]\) given by \(Y = f (X)+\varepsilon\), where \(f\) is a monotone increasing function on \([0, 1]\) and \(\varepsilon\) is a mean-zero random error with finite variance \(\sigma^2\). Therefore, it develop an easy-to-use Bayesian method for constructing credible intervals for the function value \(f(x_0)\) at an interior point \(x_0\), and obtain the asymptotic frequentist coverage of the resulting interval. Works towards the circular law conjecture have been done by many authors including Mehta, Edelman, Girko, Bai, Götze and Tikhomirov, Pan and Zhou and the present authors (see the references) under some assumptions on the distribution. For example, Mehta (complex Gaussian distribution) and Edelman (real Gaussian distribution). The authors present results on coverage of credible after introducing some notation, assumptions and the prior on \(f\) and \(\sigma\). Belongs those assumptions, we have that the isotonic regression estimator of \(f\) is obtained by minimizing the sum of squares \(\sum_i^n = (Y_i-f(X_i))^2\) subject to the constraint that \(f\) is nondecreasing on \([0, 1]\). The resulting estimator of \(f\) is a nondecreasing step function, constant on the pieces \((X_i-1,X_i]\), with \(X_0\) defined to be zero. The estimated value of \(f(x_0)\) is given by the left derivative of the greatest convex minorant of the graph of the line segments connecting \(\{(0,0), (1/n,Y_1/n), (2/n, (Y_1 +Y_2)/n),\dots , (1, (\sum_{i=1}^n Y_i)/n)\}\), at the point \(i(x_0)/n\), where \(i(x_0)\) is the integer such that \(X_{i-1} < x_0 \leq X_i\). By using the assumptions of Theorem 3.1 and Proposition 3.2 and Theorem 3.3, the authors study coverage of credible intervals for the value of the regression function at an interior point through simulations.
reviewed by: Mahamat Ali Issaka
Mathematics Subject Classification ID: 62G08
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G10
Mathematics Subject Classification ID: 62G15
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62J05
zbMATH DE Number: 7367871
zbMATH Keywords: monotonicity
zbMATH Keywords: nonparametric regression
zbMATH Keywords: credible set
zbMATH Keywords: coverage
zbMATH Keywords: Chernoff's distribution
zbMATH Keywords: projection-posterior
zbMATH Keywords: recalibration
MaRDI profile type: Publication
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full work available at URL: 20-aos1989
OpenAlex ID: W3143096306
description: scientific article; zbMATH DE number 7367871