跳跃惩罚最小二乘估计的相合性和收敛速度

Non-parametric adaptive estimation of the drift for a jump diffusion process

In this article, we consider a jump diffusion process (X_t)observed at discrete times t=0,Delta,...,nDelta. The sampling interval Delta tends to 0 and nDelta tends to infinity. We assume that (X_t) is ergodic, strictly stationary and exponentially β-mixing. We use a penalized least-square approach to compute two adaptive estimators of the drift function b. We provide bounds for the risks of the two estimators.

Non parametric estimation of the diffusion coefficents of a diffusion with jumps

In this article, we consider a jump diffusion process (X_t), with drift function b, diffusion coefficient sigma and jump coefficient xi^{2}. This process is observed at discrete times t=0,Delta,...,nDelta. The sampling interval Delta tends to 0 and nDelta tends to infinity. We assume that (X_t) is ergodic, strictly stationary and exponentially beta-mixing. We use a penalized least-square approach to compute adaptive estimators of the functions sigma^2+xi^2 and sigma^2. We provide bounds for the risks of the two estimators.

逆回归中的跳跃估计

Jump estimation in inverse regression

我们考虑从噪声观测的反卷积$ϕ*f$，其中$ϕ$是一些有界的$L_1$ -函数的阶阶梯函数$f$的估计。我们使用惩罚最小二乘估计器从观测值重建信号$f$，惩罚等于重建的跳跃数。渐近地，有可能以概率1正确估计跳跃数。鉴于跳跃的数量是正确的估计，我们表明，相应的参数估计的跳跃位置和跳跃高度是$n^{-1/2}$一致的，并收敛到一个联合的正态分布的协方差结构取决于$ϕ$，这个速度是minimax有界连续核$ϕ$。作为特例，我们得到了多相回归中最小二乘估计的渐近分布及其推广。在有界的$ϕ$得到的结果相比，我们表明，对于内核的奇异性的顺序$O(| x|^{-α}),1/2<α<1$，跳跃位置可以估计的速率为$n^{-1/(3-2α)}$，这又是极大极小速率。我们发现，这些速率不依赖于频谱信息的运营商，而是在时域中的本地化属性。最后，它证明，自适应采样不提高收敛的收敛速率，在严格的对比情况下，直接回归。

Lévy过程小跳的非参数密度估计

Nonparametric density estimation for the small jumps of Lévy processes

针对Lévy过程小跳密度无闭式解的问题，提出低频谱估计与高频极小极大自适应估计方法，移除大跳测度已知假设并推广至含布朗分量情形，数值验证了性能。

Jump Interval-Learning for Individualized Decision Making

An individualized decision rule (IDR) is a decision function that assigns each individual a given treatment based on his/her observed characteristics. Most of the existing works in the literature consider settings with binary or finitely many treatment options. In this paper, we focus on the continuous treatment setting and propose a jump interval-learning to develop an individualized interval-valued decision rule (I2DR) that maximizes the expected outcome. Unlike IDRs that recommend a single treatment, the proposed I2DR yields an interval of treatment options for each individual, making it mo…

Prediction bounds for higher order total variation regularized least squares

We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of $(k-1)^{\rm th}$ order differences. Our approach is based on combining a general oracle inequality for the $\ell_1$-penalized least squares estimator with "interpolating vectors" to upper-bound the "effective sparsity". This allows one to show that the $\ell_1$-penalty on the $k^{\text{th}}$ order differences leads to an estimator that can adapt to the number of jumps in the $(k-1)^{\text{th}}$ order differences of the underlying signal or an approximation thereof. We show the r…