部分已知噪声分布卷积模型的自适应性
Adaptivity in convolution models with partially known noise distribution
摘要：
我们考虑一个半参数卷积模型。我们观察到随机变量具有由某些未知密度$f$和某些部分已知的噪声密度$g$的卷积给出的分布。在这项工作中，假设$g$指数光滑的稳定的法律具有未知的自相似性指数$s$。为了确保模型的可识别性，我们限制我们的注意力多项式光滑，Sobolev型密度$f$，光滑参数$β$。在这种情况下，我们首先提供了一个一致的估计过程$s$。这个估计器，然后插入到三个不同的程序：估计未知密度$f$，功能$\int f^2$和拟合优度检验的假设$H_0:f=f_0$，其中替代$H_1$表示相对于$\mathbb {L}_2$规范（即具有形式$ψ_n^{-2}\|f-f_0\|_2^2\ge \mathcal{C}$）。这些过程对于$s$和$β$两者都是自适应的，并且获得对于$s$和$β$的已知值已知的最优速率。作为一个副产品，当噪声密度是已知的，指数平滑我们的测试程序是最佳的自适应测试Sobolev型密度。用合成数据说明了$s$的估计过程。
英文摘要：
We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density $f$ and some partially known noise density $g$. In this work, $g$ is assumed exponentially smooth with stable law having unknown self-similarity index $s$. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities $f$, with smoothness parameter $β$. In this context, we first provide a consistent estimation procedure for $s$. This estimator is then plugged-into three different procedures: estimation of the unknown density $f$, of the functional $\int f^2$ and goodness-of-fit test of the hypothesis $H_0:f=f_0$, where the alternative $H_1$ is expressed with respect to $\mathbb {L}_2$-norm (i.e. has the form $ψ_n^{-2}\|f-f_0\|_2^2\ge \mathcal{C}$). These procedures are adaptive with respect to both $s$ and $β$ and attain the rates which are known optimal for known values of $s$ and $β$. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of $s$ is illustrated on synthetic data.
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