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## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|DATA5441 A data-driven procedure for dependent tests in an HMM

avatest.org高维数据分析High-Dimensional Data Analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest.org™， 最高质量的高维数据分析High-Dimensional Data Analysis作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此高维数据分析High-Dimensional Data Analysis作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest.org™ 为您的留学生涯保驾护航 在商科代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的商科代考服务。我们的专家在高维数据分析High-Dimensional Data Analysis代写方面经验极为丰富，各种高维数据分析High-Dimensional Data Analysis相关的作业也就用不着 说。

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|A data-driven procedure for dependent tests in an HMM

The oracle procedure is difficult to implement since $c_{O R}$ is difficult to calculate. In addition, the HMM parameters $\vartheta$ are usually unknown. Sun and Cai (2009) derived a data-driven procedure that mimics the oralce procedure. We first estimate the unknown quantities by $\hat{\vartheta}$, then plug-in $\hat{\vartheta}$ to obtain $\mathrm{LIS}_{i}$. The maximum likelihood estimate (MLE) is commonly used and is strongly consistent and asymptotically normal under certain regularity conditions (Baum and Petrie, 1966; Leroux, 1992; Bickel et al., 1998). The MLE can be computed using the EM algorithm or other standard numerical optimization schemes, such as the gradient search, or downhill simplex algorithm. These methods are reviewed by Ephraim and Merhav (2002). In many practical applications, the number of components in the non-null mixture $L$ is unknown, yet the information is needed by the algorithms used to maximize the likelihood function. Consistent estimates of $L$ can be obtained using the method proposed by Kiefer (1993) and Liu and Narayan (1994), among others. Alternately, one can use likelihood based criteria, such as Akaike or Bayesian information criterion (BIC) to select the number of components in the normal mixture.

Let $\hat{\vartheta}$ be an estimate of the HMM parameter $\vartheta$. Define the plug-in test statistic $\mathrm{LIS}{i}(\boldsymbol{x})=P{\hat{\vartheta}}\left(\theta_{i}=0 \mid x\right)$. For given $\hat{\vartheta}, \mathrm{LIS}{i}$ can be computed via the forward-backward procedure. Denote by $\mathrm{LIS}{(1)}(\boldsymbol{x}), \ldots, \mathrm{LIS}{(m)}(\boldsymbol{x})$ the ranked plugin test statistics and $H{(1)}, \ldots, H_{(m)}$ the corresponding hypotheses. In light of the oracle procedure, we propose the following data-driven procedure:
Let $k=\max \left{i: \frac{1}{i} \sum_{j=1}^{i} \operatorname{LIS}{(j)}(x) \leqslant \alpha\right}$, then reject all $H{(i)}, i=1, \ldots, k$.
The testing procedure given in $(5.5)$ is referred to as the LIS procedure. We shall show that the performance of OR is asymptotically attained by LIS under some standard assumptions on the HMM. The asymptotic properties of the LIS procedure are studied by the following theorems. Theorem $5.3$ shows that the rejection sets yielded by OR and LIS are asymptotically equivalent in the sense that the ratio of the number of rejections and the ratio of the number of true positives yielded by the two procedures approach 1 as $m \rightarrow \infty$.

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|Simulation studies

We first assume that $L$, the number of components in non-null mixture, is known or estimated correctly from the data. The situation where $L$ is misspecified is considered in Sun and Cai (2009). In all simulations, we choose the number of hypotheses $m=3000$ and the number of replications $N=500$.

Example 5.5. The Markov chain $\left(\theta_{i}\right){1}^{m}$ is generated with the initial state distribution $\pi^{0}=\left(\pi{0}, \pi_{1}\right)=(1,0)$ and transition matrix $\mathcal{A}=\left[0.95,0.05 ; 1-a_{11}, a_{11}\right]$. The observations $\left(x_{i}\right){1}^{m}$ are generated conditional on $\left(\theta{i}\right){1}^{m}: x{i} \mid \theta_{i}=0 \sim N(0,1)$, $x_{i} \mid \theta_{i}=1 \sim N(\mu, 1)$. Figure $5.2$ compares the performance of BH, AP, OR and LIS. In the top row we choose $\mu=2$ and plot the FDR, FNR and average number of true positives (ATP) yielded by BH, AP, OR and LIS as functions of $a_{11}$. In the bottom row, we choose $a_{11}=0.8$ and plot the FDR, FNR and ATP as functions of $\mu$. The nominal FDR in all simulations is set at level $0.10$.

From Panel (a), we can see that the FDR levels of all four procedures are controlled at $0.10$ asymptotically, and the $\mathrm{BH}$ procedure is conservative. From Panels (b) and (c), we can see that the two lines of the oracle procedure and LIS procedure are almost overlapped, indicating that the performance of the oracle procedure is attained by the LIS procedure asymptotically. In addition, the two $p$-value based procedures are dominated by the LIS procedure and the difference in FNR and ATP levels becomes larger as $a_{11}$ increases. Note that $a_{11}$ is the transition probability from a non-null case to a non-null case, therefore it controls how likely the non-null cases cluster together. It is interesting to observe that the $p$-value procedures have higher FNR levels as the non-nulls cluster in larger groups. In contrast, the FNR levels of the LIS procedure decreases as $a_{11}$ increases. This observation shows that if modeled appropriately, the positive dependency is a blessing (the FNR level decreases in $a_{11}$ ); but if it is ignored, the positive dependency may become a disadvantage. In situations where the non-null cases are prevented from forming into clusters $\left(a_{11}<0.5\right)$, the LIS procedure is still more efficient than BH and AP, although the gain in efficiency is not as much as the situation where $a_{11}>0.5$.

## 商科代写|高维数据分析代考HIGHDIMENSIONAL DATA ANALYSIS代 考|A data-driven procedure for dependent tests in an HMM

(2009) 推导出了一个模仿 oracle 程序的数据驱动程序。我们首先估计末知量 $\hat{\vartheta}$, 然后揷件 $\hat{\vartheta}$ 获得LIS $L_{i}$. 最大似然估计 (MLE) 是常用的，并且在某些规律性条件下具有很强的一致性 和渐近正态性 (Baum 和 Petrie，1966；Leroux，1992；Bickel 等人，1998) 。可以 使用 EM 算法或其他标准数值优化方空 (例如梯度搜索或下坡单纯形算法) 来计算 MLE。Ephraim 和 Merhav (2002) 回顾了这些方法。在许多实际应用中，非零混合中的 成分数量 $L$ 是末知的，但用于最大化似然函数的算法需要这些信息。一致的估计 $L$ 可以使 用 Kiefer (1993) 以及 Liu 和 Narayan (1994) 等人提出的方法获得。或者，可以使用基于 可能性的标准，例如 Akaike 或贝叶斯信息标准 (BIC) 来选择正常混合中的成分数量。

: \eft 的分隔符缺失或无法识别
，然后拒绝所有 $H(i), i=1, \ldots, k$.

## 商科代写|高维数据分析代考 $\mathrm{HIGH}-$ DIMENSIONAL DATA ANALYSIS代 考|Simulation studies

$(\theta i) 1^{m}: x i\left|\theta_{i}=0 \sim N(0,1), x_{i}\right| \theta_{i}=1 \sim N(\mu, 1)$. 数字5.2比较 BH、AP、OR 和 LIS 的性能。在第一行我们选择 $\mu=2$ 并绘制由 $\mathrm{BH}$ 、AP、OR 和 LIS 产生的 FDR、FNR 和平 均真阳性数 (ATP) 作为函数 $a_{11}$. 在底行，我们选择 $a_{11}=0.8$ 并将 FDR、FNR 和 ATP 绘 制为 $\mu$. 所有模拟中的名义 FDR 均设置为水平 $0.10$.

## 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。

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## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|ETF5500 Large-scale multiple testing under dependence

avatest.org高维数据分析High-Dimensional Data Analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest.org™， 最高质量的高维数据分析High-Dimensional Data Analysis作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此高维数据分析High-Dimensional Data Analysis作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest.org™ 为您的留学生涯保驾护航 在商科代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的商科代考服务。我们的专家在高维数据分析High-Dimensional Data Analysis代写方面经验极为丰富，各种高维数据分析High-Dimensional Data Analysis相关的作业也就用不着 说。

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|Large-scale multiple testing under dependence

Observations arising from large scale multiple comparison problems are often dependent. For example, in microarray experiments, different genes may cluster into groups along biological pathways and exhibit high correlation. In public health surveillance studies, the observed data from different time periods and locations are often serially or spatially correlated. Correlation has big effects on a multiple testing procedure. Finner and Roters (2002) and Owen (2005) showed that both the expectation and variance of the number of Type I errors are greatly affected by the correlation among the hypotheses. Qiu et al. (2005) noted that the correlation effects can substantially deteriorate the performance of many $\mathrm{FDR}$ procedures. The correlation effects on the $z$-value null distribution is studied by Efron (2007), who suggested that an adjusted FDR estimate should be combined with the use of an Lfdr procedure to remove the bias caused by the correlation. Nevertheless, the works by Benjamini and Yekutieli (2001), Farcomeni (2006) and Wu (2009) show that the FDR is controlled at the nominal level by the BH step-up and adaptive $p$-value procedure under different dependence assumptions, supporting the “do nothing” approach.

Among the suggestions with respect to the correlation effects on an FDR procedure, the validity issue is overemphasized, and the efficiency issue is ignored. The FDR procedures developed under the independence assumption, even valid, may suffer from substantial efficiency loss when the dependence structure is highly informative. These situations include the geographical disease mapping studies, multiple-stage clinical trials, functional Magnetic Resonance Imaging analyses and comparative microarray experiments, where the non-null cases are often structured in some way, e.g., correlated temporally, spatially or functionally. Benjamini and Heller (2007) and Genovese et al. (2005) suggested incorporating scientific or spatial information into a multiple testing procedure to improve the efficiency. However, their approaches essentially rely on prior information, such as well defined clusters or prespecified weights, and the correlation structure among the hypotheses is not modeled.

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|The oracle procedure

Let $\delta \in{0,1}^{m}$ be a general decision rule defined as before. Sun and Cai (2009) showed that under the HMM dependency and a monotone ratio condition, the multiple testing problem is equivalent to a weighted classification problem with loss function
$$L_{\lambda}(\boldsymbol{\theta}, \boldsymbol{\delta})=\frac{1}{m} \sum_{i}\left[\lambda\left(1-\theta_{i}\right) \delta_{i}+\theta_{i}\left(1-\delta_{i}\right)\right] .$$
It can be shown that the optimal solution to the weighted classification problem is $\boldsymbol{\delta}(\boldsymbol{\Lambda}, 1 / \lambda)=\left(\delta_{1}, \ldots, \delta_{m}\right)$, where
$$\Lambda_{i}(x)=\frac{P_{\vartheta}\left(\theta_{i}=0 \mid x\right)}{P_{\vartheta}\left(\theta_{i}=1 \mid x\right)}$$

and $\delta_{i}=I\left{\Lambda_{i}(\boldsymbol{x})<1 / \lambda\right}$ for $i=1, \ldots, m$.
Remark 5.1. Given $\vartheta$, the oracle classification statistic $\Lambda_{i}(x)$ can be expressed in terms of the forward and backward density variables, which are defined as $\alpha_{i}(j)=f_{\vartheta}\left[\left(x_{t}\right){1}^{i}, \theta{i}=j\right]$ and $\beta_{i}(j)=f_{\vartheta}\left[\left(x_{t}\right){i+1}^{m} \mid \theta{i}=j\right]$, respectively (note that the dependence of $\alpha_{i}(j)$ on $\left(x_{t}\right){1}^{i}$ has been suppressed, similarly for $\left.\beta{i}(j)\right)$. It can be shown that $P_{\vartheta}\left(x, \theta_{i}=j\right)=\alpha_{i}(j) \beta_{i}(j)$ and hence $\Lambda_{i}(x)=\left[\alpha_{i}(0) \beta_{i}(0)\right] /\left[\alpha_{i}(1) \beta_{i}(1)\right]$. The forward variable $\alpha_{i}(j)$ and backward variable $\beta_{i}(j)$ can be calculated recursively using the forward-backward procedure (Baum et al. 1970 and Rabiner 1989). Specifically, we initialize $\alpha_{1}(j)=\pi_{j} f_{j}\left(x_{1}\right), \beta_{m}(j)=1$, then by induction we have $\alpha_{i+1}(j)=\left[\sum_{k=0}^{1} \alpha_{i}(k) a_{k j}\right] f_{j}\left(x_{i+1}\right)$ and $\beta_{i}(j)=\sum_{k=0}^{1} a_{j k} f_{k}\left(x_{i+1}\right) \beta_{i+1}(k)$.

Since $\Lambda_{i}(\boldsymbol{x})$ is increasing in $P_{\vartheta}\left(\theta_{i}=0 \mid x\right)$, an optimal multiple-testing rule in an HMM can be written in the form of $\delta=\left[I\left{P_{\vartheta}\left(\theta_{i}=0 \mid \boldsymbol{x}\right)<t\right}: i=1, \ldots, m\right]$. Define the local index of significance (LIS) for hypothesis $i$ by
$$\mathrm{LIS}{i}=P{\vartheta}\left(\theta_{i}=0 \mid \boldsymbol{x}\right) .$$

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|Simulation studies

$$X_{k i} \sim\left(1-p_{k}\right) N\left(\mu_{k 0}, \sigma_{k 0}^{2}\right)+p_{k} N\left(\mu_{k}, \sigma_{k}^{2}\right), k=1,2 .$$
PLfdr、SLfdr 和 CLfdr 程序的数值性能将在接下来的模拟研究中进行研究。名义上的全 球 FDR 水平是0.10.

$0.9 N(0,1)+0.1 N(4,1)$. 我们不一样 $p_{1}$ ，第 1 组中非空值的比例，并将 FDR 和 FNR 水 平绘制为 $p_{1}$. (ii) 组的大小也是 $m_{1}=3000$ 和 $m_{2}=1500$; 组混合pdf是
$f_{1}=0.8 N(0,1)+0.2 N\left(\mu_{1}, 1\right)$ 和 $f_{2}=0.9 N(0,1)+0.1 N\left(2,0.5^{2}\right)$. FDR 和 FNR 水平 绘制为 $\mu_{1}$. (iii) 边际 pdf 是 $f_{1}=0.8 N(0,1)+0.2 N\left(-2,0.5^{2}\right)$ 和
$f_{2}=0.9 N(0,1)+0.1 N(4,1)$. 第 2 组的样本量固定为 $m_{2}=1500$, FDR 和 FNR 水平被 绘制为函数 $m_{1}$. 图 $4.3$ 给出了 500 次重复的模拟结果。第一行比较了三个程序的实际 FDR 水平；设置 (i)、(ii) 和 (iii) 的结果分别显示在面板 (a)、(b) 和 (c) 中。还提供了
CLfdr 程序的分组 FDR 水平（第 1 组的虚线和第 2 组的虚线）。最下面一行比较了三个 程序的 FNR 水平；设置 (i)、(ii) 和 (iii) 的结果分别显示在面板 (d)、 (e) 和 (f) 中。

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

AYP 研究的一个目标是比较社会经济优势 (SEA) 与社会经济嵶势 (SED) 学生的数学考试 成功率。由于 SEA 学生的平均成功率通常 (7867 所学校中的 7370 所) 高于 SED 学 生，因此有必要确定一个学校的子集，在这些学校中，优势与劣势的表现差异异常小或 大。表示为 $X_{i}$ 和 $Y_{i}$ 成功率，以及 $n_{i}$ 和 $n_{i}^{\prime}$ 在校 SEA 和 SED 学生报告的分数数量 $i$,
$i=1, \ldots, m$. 定义居中常数 $\Delta=\operatorname{median}\left(X_{i}\right)-\operatorname{median}\left(Y_{i}\right) .$ 一个 $z$ 可以为每所学校计算 比较 SEA 学生与 SED 学生的值:
$$z_{i}=\frac{X_{i}-Y_{i}-\Delta}{\sqrt{X_{i}\left(1-X_{i}\right) / n_{i}+Y_{i}\left(1-Y_{i}\right) / n_{i}^{\prime}}},$$

## 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。

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## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|ETF3500 Simulation studies

avatest.org高维数据分析High-Dimensional Data Analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest.org™， 最高质量的高维数据分析High-Dimensional Data Analysis作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此高维数据分析High-Dimensional Data Analysis作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest.org™ 为您的留学生涯保驾护航 在商科代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的商科代考服务。我们的专家在高维数据分析High-Dimensional Data Analysis代写方面经验极为丰富，各种高维数据分析High-Dimensional Data Analysis相关的作业也就用不着 说。

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|Simulation studies

Consider the following two-group normal mixture model:
$$X_{k i} \sim\left(1-p_{k}\right) N\left(\mu_{k 0}, \sigma_{k 0}^{2}\right)+p_{k} N\left(\mu_{k}, \sigma_{k}^{2}\right), k=1,2 .$$
The numerical performances of the PLfdr, SLfdr and CLfdr procedures are investigated in the next simulation study. The nominal global FDR level is $0.10$.

Example 4.5. The null distributions of both groups are fixed as $N(0,1)$. Three simulation settings are considered: (i) The group sizes are $m_{1}=3000$ and $m_{2}=$ 1500 ; the group mixture pdf’s are $f_{1}=\left(1-p_{1}\right) N(0,1)+p_{1} N(-2,1)$ and $f_{2}=$ $0.9 N(0,1)+0.1 N(4,1)$. We vary $p_{1}$, the proportion of non-nulls in group 1 , and plot the FDR and FNR levels as functions of $p_{1}$. (ii) The groups sizes are also $m_{1}=$ 3000 and $m_{2}=1500$; the group mixture pdf’s are $f_{1}=0.8 N(0,1)+0.2 N\left(\mu_{1}, 1\right)$ and $f_{2}=0.9 N(0,1)+0.1 N\left(2,0.5^{2}\right)$. The FDR and FNR levels are plotted as functions of $\mu_{1}$. (iii) The marginal pdf’s are $f_{1}=0.8 N(0,1)+0.2 N\left(-2,0.5^{2}\right)$ and $f_{2}=0.9 N(0,1)+0.1 N(4,1)$. The sample size of group 2 is fixed at $m_{2}=1500$, the FDR and FNR levels are plotted as functions of $m_{1}$. The simulation results with 500 replications are given in Figure 4.3. The top row compares the actual FDR levels of the three procedures; the results for setting (i), (ii) and (iii) are shown in Panels (a), (b) and (c), respectively. The group-wise FDR levels of the CLfdr procedure are also provided (the dashed line for group 1 and dotted line for group 2). The bottom row compares the FNR levels of the three procedures; the results for setting (i), (ii) and (iii) are shown in Panels (d), (e) and (f), respectively.
We can see that all three procedures control the global FDR level at the nominal level $0.10$, indicating that all three procedures are valid. It is important to note that the CLfdr procedure chooses group-wise FDR levels automatically (dashed and dotted lines in Panels (a)-(c)), and the levels are in general different from the nominal level 0.10. The relative efficiency of PLfdr versus SLfdr is inconclusive (depends on simulation settings). For example, the SLfdr procedure yields lower FNR levels in Panel (d), but higher FNR levels in Panel (f). However, all simulations show that both the PLfdr and SLfdr procedures are uniformly dominated by the CLfdr procedure.

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

We now return to the adequate yearly progress (AYP) study mentioned in the introduction. In this section, we analyze the data collected from $m=7867$ of California high schools (Rogosa 2003) by using the PLfdr, SLfdr and CLfdr procedures.

One goal of the AYP study is to compare the success rates in Math exams of social-economically advantaged (SEA) versus social-economically disadvantaged (SED) students. Since the average success rates of the SEA students are in general ( 7370 out of 7867 schools) higher that the SED students, it is of interest to identify a subset of schools in which the advantaged-disadvantaged performance differences are unusually small or large. Denote by $X_{i}$ and $Y_{i}$ the success rates, and $n_{i}$ and $n_{i}^{\prime}$ the numbers of scores reported for SEA and SED students in school $i$,

$i=1, \ldots, m$. Define the centering constant $\Delta=\operatorname{median}\left(X_{i}\right)-\operatorname{median}\left(Y_{i}\right)$. A $z$ value for comparing the SEA students versus the SED students can be computed for each school:
$$z_{i}=\frac{X_{i}-Y_{i}-\Delta}{\sqrt{X_{i}\left(1-X_{i}\right) / n_{i}+Y_{i}\left(1-Y_{i}\right) / n_{i}^{\prime}}},$$
for $i=1, \ldots, m$. We claim school $i$ is “interesting” if the observed $\left|z_{i}\right|$ is large.

## 商科代写|高维数据分析代考HIGH-DIMENSIONAL DATA ANALYSIS代考|Simulation studies

$$X_{k i} \sim\left(1-p_{k}\right) N\left(\mu_{k 0}, \sigma_{k 0}^{2}\right)+p_{k} N\left(\mu_{k}, \sigma_{k}^{2}\right), k=1,2 .$$
PLfdr、SLfdr 和 CLfdr 程序的数值性能将在接下来的模拟研究中进行研究。名义上的全 球 FDR 水平是0.10.

$0.9 N(0,1)+0.1 N(4,1)$. 我们不一样 $p_{1}$ ，第 1 组中非空值的比例，并将 FDR 和 FNR 水 平绘制为 $p_{1}$. (ii) 组的大小也是 $m_{1}=3000$ 和 $m_{2}=1500$; 组混合pdf是
$f_{1}=0.8 N(0,1)+0.2 N\left(\mu_{1}, 1\right)$ 和 $f_{2}=0.9 N(0,1)+0.1 N\left(2,0.5^{2}\right)$. FDR 和 FNR 水平 绘制为 $\mu_{1}$. (iii) 边际 pdf 是 $f_{1}=0.8 N(0,1)+0.2 N\left(-2,0.5^{2}\right)$ 和
$f_{2}=0.9 N(0,1)+0.1 N(4,1)$. 第 2 组的样本量固定为 $m_{2}=1500$, FDR 和 FNR 水平被 绘制为函数 $m_{1}$. 图 $4.3$ 给出了 500 次重复的模拟结果。第一行比较了三个程序的实际 FDR 水平；设置 (i)、(ii) 和 (iii) 的结果分别显示在面板 (a)、(b) 和 (c) 中。还提供了
CLfdr 程序的分组 FDR 水平（第 1 组的虚线和第 2 组的虚线）。最下面一行比较了三个 程序的 FNR 水平；设置 (i)、(ii) 和 (iii) 的结果分别显示在面板 (d)、 (e) 和 (f) 中。

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

AYP 研究的一个目标是比较社会经济优势 (SEA) 与社会经济嵶势 (SED) 学生的数学考试 成功率。由于 SEA 学生的平均成功率通常 (7867 所学校中的 7370 所) 高于 SED 学 生，因此有必要确定一个学校的子集，在这些学校中，优势与劣势的表现差异异常小或 大。表示为 $X_{i}$ 和 $Y_{i}$ 成功率，以及 $n_{i}$ 和 $n_{i}^{\prime}$ 在校 SEA 和 SED 学生报告的分数数量 $i$,
$i=1, \ldots, m$. 定义居中常数 $\Delta=\operatorname{median}\left(X_{i}\right)-\operatorname{median}\left(Y_{i}\right) .$ 一个 $z$ 可以为每所学校计算 比较 SEA 学生与 SED 学生的值:
$$z_{i}=\frac{X_{i}-Y_{i}-\Delta}{\sqrt{X_{i}\left(1-X_{i}\right) / n_{i}+Y_{i}\left(1-Y_{i}\right) / n_{i}^{\prime}}},$$

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