噪声分布已知或部分已知的卷积模型中的自适应过程
Adaptive procedures in convolution models with known or partially known noise distribution
摘要：
在卷积模型中，我们观察随机变量，其分布是某些未知密度f和某些已知或部分已知的噪声密度g的卷积。在本文中，我们专注于统计程序，这是自适应的光滑参数τ的未知密度f，也（在某些情况下）的一些未知参数的噪声密度g。在第一部分中，我们假设g是已知的且多项式光滑的。本文给出了检验H_0：f=f_0的拟合优度方法，其中H_1是关于L_2-范数表示的。我们的自适应（w.r.t tau）程序根据f_0是多项式光滑还是指数光滑而表现不同。支付适应注意到在这两种情况下，计算这一点，我们提供了一个非均匀的Berry-Esseen型定理退化U-统计。在第一种情况下，我们证明了适应的支付是最优的（因此是不可避免的）。在第二部分中，我们研究了一个更广泛的框架：半参数模型，其中g是指数光滑和稳定的，其自相似指数s是未知的。为了确保可识别性，我们将注意力限制在多项式光滑的Sobolev型密度f上。在这种情况下，我们提供了一个一致的估计程序的s。这个估计器被插入到三个不同的过程中：未知密度f的估计，泛函\int f^2的估计和假设H_0的检验。这些过程对于s和tau都是自适应的，并且获得对于已知的s和tau值已知的最优速率。作为一个副产品，当噪声是已知的，指数平滑的测试程序是自适应测试Sobolev型密度。
英文摘要：
In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known or partially known noise density g. In this paper, we focus on statistical procedures, which are adaptive with respect to the smoothness parameter tau of unknown density f, and also (in some cases) to some unknown parameter of the noise density g. In a first part, we assume that g is known and polynomially smooth. We provide goodness-of-fit procedures for the test H_0:f=f_0, where the alternative H_1 is expressed with respect to L_2-norm. Our adaptive (w.r.t tau) procedure behaves differently according to whether f_0 is polynomially or exponentially smooth. A payment for adaptation is noted in both cases and for computing this, we provide a non-uniform Berry-Esseen type theorem for degenerate U-statistics. In the first case we prove that the payment for adaptation is optimal (thus unavoidable). In a second part, we study a wider framework: a semiparametric model, where g is exponentially smooth and stable, and its self-similarity index s is unknown. In order to ensure identifiability, we restrict our attention to polynomially smooth, Sobolev-type densities f. In this context, we 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 test of the hypothesis H_0. These procedures are adaptive with respect to both s and tau and attain the rates which are known optimal for known values of s and tau. As a by-product, when the noise is known and exponentially smooth our testing procedure is adaptive for testing Sobolev-type densities.
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