Semiparametric analysis of binary response from response-based samples
type: Item
instance of: scholarly article
title: Semiparametric analysis of binary response from response-based samples
zbMATH Open document ID: 0616.62146
DOI: 10.1016/0304-4076(86)90054-0
author: Charles F. Manski
published in: Journal of Econometrics
publication date: 1986-01-01T00:00:00Z
review text: Let (y,x) be a random vector ranging over a subset of \({\mathbb{R}}\times {\mathbb{R}}^ K\). Let P be the \((1+K)\)-variate probability measure. Let \(P_{y| x}\) be the univariate probability measure of y conditional on x and let \(P_ x\) be the K-variate marginal probability measure of x. For a given \(\alpha\in (0,1)\), let \(Q_{\alpha}(y| x)\) denote the \(\alpha\)-quantile of y conditional on x. We assume that there exists a unique \(\beta \in {\mathbb{R}}^ K\) such that \(Q_{\alpha}(y| x)=x\beta\). In part of the analysis, we assume that for all \(\alpha\in (0,1)\), the \(\alpha\)-quantile regressions are parallel. We impose no further restrictions on \(P_{y| x}\). This semiparametric setting contrasts with parametric frameworks where \(P_{y| x}\) is restricted to a smooth finite-dimensional family of conditional measures. Section 2 shows that if the marginal response probabilities are known, an appropriately weighted version of the maximum score estimator can be used to estimate \(\beta^*\) consistently where \(\beta^*=\beta /\| \beta \|\) for a norm \(\| \cdot \|\) on \({\mathbb{R}}^ K\). The weighted estimator has an equivalent representation as a shifted maximum score estimator. Section 3 shows that in the absence of information on marginal probabilities, a strengthening of the assumptions on conditional probabilities permits inference on \(\beta\). In particular, if all the quantile regressions of y on x are known to be parallel, a weighted- shifted maximum score estimator can be used to estimate the slope components of \(\beta^*\) consistently. With these positive findings in hand, it is natural to ask what features of \(\beta^*\) can be estimated from response-based samples in the absence of any information beyond the basic assumption of a linear quantile regression. This question remains unanswered.
Mathematics Subject Classification ID: 62P20
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G99
zbMATH DE Number: 3999093
zbMATH Keywords: semiparametric analysis
zbMATH Keywords: binary response
zbMATH Keywords: semiparametric estimation
zbMATH Keywords: Glivenko-Cantelli type uniform law of large numbers
zbMATH Keywords: strong consistency
zbMATH Keywords: maximum score estimator
zbMATH Keywords: marginal probabilities
zbMATH Keywords: response-based samples
zbMATH Keywords: linear quantile regression
MaRDI profile type: Publication
full work available at URL: 0304-4076(86)90054-0
OpenAlex ID: W2045804992
cites work: Estimation of Response Probabilities From Augmented Retrospective Observations
cites work: Maximum score estimation of the stochastic utility model of choice
cites work: Closest Empirical Distribution Estimation
cites work: Semiparametric analysis of discrete response. Asymptotic properties of the maximum score estimator
cites work: The Estimation of Choice Probabilities from Choice Based Samples
cites work: Q5185819
cites work: Relations between Weak and Uniform Convergence of Measures with Applications
description: scientific article; zbMATH DE number 3999093