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Title: Global testing under sparse alternative for single index model Authors:  Qian Lin - Tsinghua University (China) [presenting]
Abstract: Testing for the significance of a signal in a linear model goes back at least to the work of Fisher. We study this problem for the single index model $y=f(\beta^{\tau}\boldsymbol{x},\epsilon)$ with Gaussian design where $f$ is unknown and $\beta$ is a $p$ dimensional unit vector with at most $s$ nonzero entries. We adopt the notion of generalized signal to noise ratio (gSNR). We are interested in the hypothesis testing problem of whether $\beta=0$ or not. Let $n$ be the size of observed data. We show that if $s^{2}\wedge p\prec n$, one can detect the gSNR if and only if $gSNR\succ\frac{p^{1/2}}{n}\wedge \frac{s\log(p)}{n}$. Furthermore, if the noise is additive( i.e., $y=f(\beta^{\tau}\boldsymbol{x})+\epsilon$), one can detect gSNR if and only if $gSNR\succ\frac{p^{1/2}}{n}\wedge \frac{s\log(p)}{n} \wedge \frac{1}{\sqrt{n}}$. In other words, the detection boundary gSNR for the single index model with additive noise matches that of SNR for linear regression. These results pave the road of a through treatment of single/multiple index models in high dimensions. For example, one may try to extend the well developed theories of linear models to the single/multiple index models with Gaussian design.