Nonparametric estimation of the derivatives of the stationary density for stationary processes
Nonparametric estimation of the derivatives of the stationary density for stationary processes
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平稳过程平稳密度导数的非参数估计
DOI：
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发表时间：
2013
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通讯作者：
Emeline Schmisser
中科院分区：
文献类型：
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作者：
Emeline Schmisser
本文的目的是估计一个严格平稳β -混合过程（Xt）t≥0的平稳密度f的连续导数。在离散时间t = 0，Δ，.，nΔ .采样间隔Δ可以是固定的或小的。我们使用惩罚最小二乘方法来计算自适应估计。如果导数f（j）属于Besov空间，则我们的估计以速率（nΔ）−α /（2α +2j +1）收敛。然后我们考虑一个扩散系数已知的扩散。我们使用平稳密度的特殊形式来计算其一阶导数f ′的自适应估计。当采样间隔Δ趋于0时，并且当扩散系数已知时，我们的估计的收敛速度是（nΔ）-α /（2α +1）。当扩散系数已知时，我们还构造了低频数据漂移的商估计。
In this article, our aim is to estimate the successive derivatives of the stationary density f of a strictly stationary and β -mixing process (Xt )t≥0 . This process is observed at discrete times t = 0,Δ, ... ,nΔ . The sampling interval Δ can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative f (j ) belongs to the Besov space , then our estimator converges at rate (nΔ )−α /(2α +2j +1) . Then we consider a diffusion with known diffusion coefficient. We use the particular form of the stationary density to compute an adaptive estimator of its first derivative f ′. When the sampling interval Δ tends to 0, and when the diffusion coefficient is known, the convergence rate of our estimator is (nΔ )−α /(2α +1) . When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data.