Estimation and convergence rates in the distributional single index model
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911761553162240 |
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| author | Balabdaoui, Fadoua Henzi, Alexander Looser, Lukas |
| author_facet | Balabdaoui, Fadoua Henzi, Alexander Looser, Lukas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13973 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Estimation and convergence rates in the distributional single index model Balabdaoui, Fadoua Henzi, Alexander Looser, Lukas Statistics Theory Methodology 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. |
| title | Estimation and convergence rates in the distributional single index model |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2310.13973 |