Estimation and convergence rates in the distributional single index model

Abstract

The distributional single index model is a semiparametric regression model in which the conditional distribution functions $P(Y \leq y | X = x) = F_0(θ_0(x), y)$ of a real-valued outcome variable $Y$ depend on $d$-dimensional covariates $X$ through a univariate, parametric index function $θ_0(x)$, and increase stochastically as $θ_0(x)$ increases. We propose least squares approaches for the joint estimation of $θ_0$ and $F_0$ in the important case where $θ_0(x) = α_0^{\top}x$ and obtain convergence rates of $n^{-1/3}$, thereby improving an existing result that gives a rate of $n^{-1/6}$. A simulation study indicates that the convergence rate for the estimation of $α_0$ might be faster. Furthermore, we illustrate our methods in a real data application that demonstrates the advantages of shape restrictions in single index models.

Figures (1)

• Figure 1: Pairs $(\hat{\alpha}_n^{\top}X_i, Y_i)$, $i = 1, \dots, 414$, for the distributional single index model, the monotone single index model, and for non-crossing quantile regression. The lines for the distributional methods are estimated conditional quantile curves at the levels $\tau = 0.1, 0.5, 0.9$.

Theorems & Definitions (20)

• Proposition 1
• Theorem 1
• Proposition 2
• Remark 1
• Theorem 2
• Remark 2
• Theorem 3
• Lemma 1
• Remark 3
• proof : Proof of Theorem \ref{['thm:bundled']}