标题： 逆回归中的跳跃估计 显示英文标题

标题： Jump estimation in inverse regression

摘要： 我们考虑从退卷积 $\phi*f$ 的噪声观测值估计一个跃阶函数 $f$，其中 $\phi$ 是某个有界的 $L_1$- 函数。 我们使用惩罚最小二乘估计器从观测值重构信号 $f$，惩罚项等于重构的跳跃点个数。 渐近地，以概率 1 可以正确估计跳跃点的个数。 假设跳跃次数被正确估计，我们证明相应的跳跃位置和跳跃高度参数估计量是$n^{-1/2}$相合的，并且收敛到一个联合正态分布，其协方差结构依赖于$\phi$，并且该收敛速度是最优的（minimax）对于有界连续核$\phi$。 作为特殊情况，我们得到了多相回归中最小二乘估计量的渐近分布及其推广形式。 与对有界核$\phi$所得的结果相反，我们证明了对于具有$O(| x|^{-\alpha}),1/2<\alpha<1$阶奇异性核的情况，跳跃位置可以以$n^{-1/(3-2\alpha)}$的速率被估计，这同样是最优的（minimax）。 我们发现这些收敛速度并不依赖于算子的谱信息，而是依赖于它在时域中的局部化性质。 最后，结果表明自适应采样并不能提高收敛速度，这与直接回归的情况形成了鲜明对比。 显示英文摘要

摘要： We consider estimation of a step function $f$ from noisy observations of a deconvolution $\phi*f$, where $\phi$ is some bounded $L_1$-function. We use a penalized least squares estimator to reconstruct the signal $f$ from the observations, with penalty equal to the number of jumps of the reconstruction. Asymptotically, it is possible to correctly estimate the number of jumps with probability one. Given that the number of jumps is correctly estimated, we show that the corresponding parameter estimates of the jump locations and jump heights are $n^{-1/2}$ consistent and converge to a joint normal distribution with covariance structure depending on $\phi$, and that this rate is minimax for bounded continuous kernels $\phi$. As special case we obtain the asymptotic distribution of the least squares estimator in multiphase regression and generalisations thereof. In contrast to the results obtained for bounded $\phi$, we show that for kernels with a singularity of order $O(| x|^{-\alpha}),1/2<\alpha<1$, a jump location can be estimated at a rate of $n^{-1/(3-2\alpha)}$, which is again the minimax rate. We find that these rate do not depend on the spectral information of the operator rather on its localization properties in the time domain. Finally, it turns out that adaptive sampling does not improve the rate of convergence, in strict contrast to the case of direct regression.