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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

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## 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 $0.10$.

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