This study is devoted to the use of wavelets in two different fields, constructions of bases on the interval and long-memory parameter estimation. In the first part we present general constructions of orthogonal and biorthogonal multiresolution analyses on the interval. In the first one, we describe a direct method to define an orthonormal multiresolution analysis. In the second one, we use the integration and derivation method for constructing a biorthogonal multiresolution analysis. As applications, we prove that these analyses are adapted to study regular functions on the interval (H^{s}([0,1])$ et $H^{s}_{0}([0,1])$ for $sinmathbb{N}$). The second part is devoted to the study of adaptive wavelet-based estimators of the long-memory parameter for Gaussian then linear processes in a general semiparametric frame. We introduce and develop the choice of a data-driven optimal bandwidth. Moreover, we establish central limit theorems for the estimators of the memory parameter with the minimax rate of convergence (up to a logarithm factor). Finally adaptive goodness-of-fit tests are also built and easy to be employed: they are chi-square type tests. Simulations confirm the interesting properties of consistency and robustness of the adaptive estimators and tests.
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机译:这项研究致力于小波在两个不同领域中的使用,即基于区间的构造和长记忆参数估计。在第一部分中,我们介绍了区间上正交和双正交多分辨率分析的一般构造。在第一个中,我们描述了定义正交多分辨率分析的直接方法。在第二篇文章中,我们使用积分和推导方法构建双正交多分辨率分析。作为应用,我们证明这些分析适用于研究区间(H ^ {s}([0,1])$和$ H ^ {s} _ {0}([0,1])$为$ s in mathbb {N} $)。第二部分致力于研究一般半参数框架中高斯然后线性过程的长记忆参数的自适应小波估计器。我们介绍并开发数据驱动的最佳带宽的选择。此外,我们建立了具有最小最大收敛速度(不超过对数因子)的内存参数估计量的中心极限定理。最后,还建立了自适应拟合优度测试,并且易于采用:它们是卡方检验。仿真证实了自适应估计器和测试的一致性和鲁棒性的有趣特性。
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