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# 商科代写|高维数据分析代考High-Dimensional Data Analysis代考|INFS4205 Oracle testing procedure

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## 商科代写|高维数据分析代考High-Dimensional Data Analysis代考|Oracle testing procedure

We have shown that $\delta^{\lambda}(\Lambda, 1 / \lambda)=\left[I\left{\Lambda\left(x_{1}\right)<1 / \lambda\right}, \ldots, I\left{\Lambda\left(x_{m}\right)<1 / \lambda\right}\right]$ is the oracle rule in the weighted classification problem. The equivalence between multiple testing and weighted classification implies the optimal testing rule is also of the form $\delta^{\lambda(\alpha)}[\Lambda, 1 / \lambda(\alpha)]$ if $\Lambda \in \mathcal{T}$, although the cutoff $1 / \lambda(\alpha)$ is not obvious. Note that $\Lambda(x)=\operatorname{Lfdr}(x) /[1-\operatorname{Lfdr}(x)]$ is monotonically increasing in $\operatorname{Lfdr}(x)$, where $\operatorname{Lfdr}(\cdot)=(1-p) f_{0}(\cdot) / f(\cdot)$ is the local false discovery rate (Lfdr) introduced by Efron et al. (2001) and Efron (2004), so the optimal rule for mFDR control is of the form $\delta(\operatorname{Lfdr}(\cdot), c)=\left{I\left[\operatorname{Lfdr}\left(x_{i}\right)<c\right]: i=1, \ldots, m\right}$. The Lfdr has been widely used in the FDR literature to provide a Bayesian version of the frequentist FDR measure and interpret results for individual cases (Efron 2004). We rediscover it here as the optimal (oracle) statistic in the multiple testing problem in the sense that the thresholding rule based on $\operatorname{Lfdr}(X)$ controls the mFDR at the nominal level with the smallest mFNR.

The MRC implies that in order to minimize the mFNR level, we should choose the largest threshold for the Lfdr statistic. Therefore the oracle testing procedure is
$$\delta\left(\operatorname{Lfdr}, c_{O R}\right)=\left{I\left[\operatorname{Lfdr}\left(x_{i}\right)<c_{O R}\right]: i=1, \ldots, m\right},$$
where the oracle threshold $c_{O R}=\sup {c \in(0,1): \operatorname{mFDR}(c) \leqslant \alpha}$. The oracle procedure (3.13) provides an ideal target for evaluating different multiple testing procedures. In particular, it is more efficient than the $p$-value oracle procedure proposed in Genovese and Wasserman (2002). Hence the $z$-value oracle procedure is more efficient than all $p$-value based FDR procedures.

## 商科代写|高维数据分析代考High-Dimensional Data Analysis代考|A data-driven procedure

The oracle procedure is not applicable in practice because the distributional information is usually unknown. This section first discusses the estimation of the null distribution and the non-null proportion in large-scale multiple comparisons. Then we introduce a data-driven procedure that mimics the oracle procedure.
Efron (2004) raised an important issue that in many large-scale studies the usual assumption that the null distribution is known is incorrect, and seemingly negligible differences in the null may result in large differences in subsequent studies. It was demonstrated that the null distribution should be estimated from data instead of being assumed known. Besides the null distribution, the proportion of non-null effects $p$ is also an important quantity. The implementation of many FDR procedures requires the knowledge of $p$ (BH 2000; Storey 2002; GW 2004). Developing good estimators for the proportion of non-nulls is a challenging task. Recent work includes that of Genovese and Wasserman (2004), Langaas, Lindqvist and Ferkingstad (2005), Meinshausen and Rice (2006), Cai, Jin and Low (2007), and Jin and Cai (2007).

Jin and Cai (2007) developed an approach based on the empirical characteristic function and Fourier analysis for simultaneous estimation of both the null distribution $f_{0}$ and proportion of non-null effects $p$. The estimators are shown to be uniformly consistent over a wide class of parameters. Numerical results also showed that the estimators perform favorably in comparison to other existing methods. This method will be used in our data-driven procedure.

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|ORACLE TESTING PROCEDURE

MRC 意味着为了最小化 mFNR 水平，我们应该选择 Lfdr统计量的最大阈值。因此，oracle 测试程序是

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|A DATA-DRIVEN PROCEDURE

$B H 2000 ;$ Storey $2002 ; G W 2004$. 为非空值的比例开发良好的估计器是一项具有挑战侏的任穷。最近的工作包括 Genovese 和 Wasserman2004, Langaas, Lindquist 和
Jin and Cai 2007开发了一种基于经验特征函数和傅里叶分析的方法，用于同时估计零分布 $f_{0}$ 和非零效应的比例 $p$. 估计量被证明在广泛的参数熠别中是一致的。数值结

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。