Semiparametric analysis of binary response from response-based samples
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.
Glivenko-Cantelli type uniform law of large numbers
scientific article; zbMATH DE number 3999093