# 英国补考|统计推断代写Statistical inference代考|STAT3013

#### Doug I. Jones

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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## 英国补考|统计推断代写Statistical inference代考|The construction

Let us associate with $i \leq M$ and $j, 0 \leq j \leq N$, the hypothesis $H_{i j}$ stating that the observations $\omega_{k}$ independent across $k$-see (2.85) -stem from
$$x_{*} \in X_{i j}:=\left{x \in X:\left|x-x_{i}\right| \leq r_{j}\right} .$$
Note that the sets $X_{i j}$ are convex and compact. We denote by $\mathcal{J}$ the set of all pairs $(i, j)$, for which $i \in{1, \ldots, M}, j \in{0,1, \ldots, N}$, and $X_{i j} \neq \emptyset$. Further, we define closeness $\mathcal{C}{\alpha, \beta}$ on the set of hypotheses $H{i j},(i, j) \in \mathcal{J}$, as follows:
$\left(i j, i^{\prime} j^{\prime}\right) \in \mathcal{C}{\alpha \beta}$ if and only if $$\left|x{i}-x_{i^{\prime}}\right| \leq \bar{\alpha}\left(r_{j}+r_{j^{\prime}}\right)+\beta, \bar{\alpha}=\frac{\alpha-1}{2}$$
(here and in what follows, $k \ell$ denotes the ordered pair $(k, \ell)$ ).
Applying Theorem 2.23, we can build, in a computation-friendly fashion, the system $\phi_{i j, i^{\prime} j^{\prime}}(\omega), i j, i^{\prime} j^{\prime} \in \mathcal{J}$, of optimal balanced detectors for the hypotheses $H_{i j}$ along with the risks of these detectors, so that
$$\begin{array}{ll} \phi_{i j, i^{\prime} j^{\prime}}(\omega) \equiv-\phi_{i^{\prime} j^{\prime}, i j}(\omega) & \forall\left(i j, i^{\prime} j^{\prime} \in \mathcal{J}\right) \ \int_{\Omega} \mathrm{e}^{-\phi_{i j, i^{\prime} j^{\prime}}(\omega)} p_{A(x)}(\omega) \Pi(d \omega) \leq \epsilon_{i j, i^{\prime} j^{\prime}} & \forall\left(i j \in \mathcal{J}, i^{\prime} j^{\prime} \in \mathcal{J}, x \in X_{i j}\right) . \end{array}$$
Let us say that a pair $(\alpha, \beta)$ is admissible if $\alpha \geq 1, \beta \geq 0$, and
$$\forall\left((i, j) \in \mathcal{J},\left(i^{\prime}, j^{\prime}\right) \in \mathcal{J},\left(i j, i^{\prime} j^{\prime}\right) \notin \mathcal{C}{\alpha, \beta}\right): A\left(X{i j}\right) \cap A\left(X_{i^{\prime} j^{\prime}}\right)=\emptyset .$$
Note that checking admissibility of a given pair $(\alpha, \beta)$ is a computationally tractable task.

## 英国补考|统计推断代写Statistical inference代考|A modification

From the computational viewpoint, an obvious shortcoming of the construction presented in the previous section is the necessity to nperate with $M(N+1)$ hypotheses, which might require computing as many as $O\left(M^{2} N^{2}\right)$ detectors. We are about to present a modified construction, where we deal at most $N+1$ times with just $M$ hypotheses at a time (i.e., with the total of at most $O\left(M^{2} N\right)$ detectors). The idea is to replace simultaneously processing all hypotheses $H_{i j}, i j \in \mathcal{J}$, with processing them in stages $j=0,1, \ldots$, with stage $j$ operating only with the hypotheses $H_{i j}$, $i=1, \ldots, M$.

The implementation of this idea is as follows. In the situation of Section $2.5 .3$, given the same entities $\Gamma,(\alpha, \beta), H_{i j}, X_{i j}, i j \in \mathcal{J}$, as at the beginning of Section 2.5.3.1 and specifying closeness $\mathcal{C}_{\alpha, \beta}$ according to (2.87), we now act as follows.
Preprocessing. For $j=0,1, \ldots, N$

1. we identify the set $\mathcal{I}{j}=\left{i \leq M: X{i j} \neq \emptyset\right}$ and stop if this set is empty. If this set is nonempty,
2. we specify the closeness $\mathcal{C}{\alpha \beta}^{j}$ on the set of hypotheses $H{i j}, i \in \mathcal{I}{j}$, as a “slice” of the closeness $C{\alpha, \beta}$ :
$H_{i j}$ and $H_{i^{\prime} j}$ (equivalently, $i$ and $\left.i^{\prime}\right)$ are $\mathcal{C}{\alpha, \beta^{j}}$-close to each other if $\left(i j, i^{\prime} j\right)$ are $\mathcal{C}{\alpha, \beta}$-close, that is,
$$\left|x_{i}-x_{i^{\prime}}\right| \leq 2 \bar{\alpha} r_{j}+\beta, \bar{\alpha}=\frac{\alpha-1}{2} .$$
3. We build the optimal detectors $\phi_{i j, i^{\prime} j}$, along with their risks $\epsilon_{i j, i^{\prime} j}$, for all $i, i^{\prime} \in \mathcal{I}{j}$ such that $\left(i, i^{\prime}\right) \notin \mathcal{C}{\alpha, \beta}^{j}$. If $\epsilon_{i j, i^{\prime} j}=1$ for a pair $i, i^{\prime}$ of the latter type, that is,
4. $A\left(X_{i j}\right) \cap A\left(X_{i^{\prime} j}\right) \neq \emptyset$, we claim that $(\alpha, \beta)$ is inadmissible and stop. Otherwise we find the smallest $K=K_{j}$ such that the spectral norm of the symmetric $M \times M$ matrix $E^{j K}$ with the entries
5. $$6. E_{i i^{\prime}}^{j K}= \begin{cases}\epsilon_{i j, i^{\prime} j}^{K}, & i \in \mathcal{I}{j}, i^{\prime} \in \mathcal{I}{j},\left(i, i^{\prime}\right) \notin \mathcal{C}{\alpha, \beta}^{j} \ 0, & \text { otherwise }\end{cases}$$ does not exceed $\bar{\epsilon}=\epsilon /(N+1)$. We then use the machinery of Section $2.5 .2 .3$ to build detector-based test $\mathcal{T}{\mathcal{C}{\alpha, \beta}^{j}}^{K{j}}$, which decides on the hypotheses $H_{i j}, i \in \mathcal{I}{j}$, with $\mathcal{C}{\alpha, \beta^{j}}$-risk not exceeding $\bar{\epsilon}$.

# 统计推断代考

## 英国补考|统计推断代写Statistical inference代考|The construction

stem from

$H i j,(i, j) \in \mathcal{J}$ ，如下:
$\left(i j, i^{\prime} j^{\prime}\right) \in \mathcal{C} \alpha \beta$ 当且仅当
$$\left|x i-x_{i^{\prime}}\right| \leq \bar{\alpha}\left(r_{j}+r_{j^{\prime}}\right)+\beta, \bar{\alpha}=\frac{\alpha-1}{2}$$
(这里和下文中， $k \ell$ 表示有序对 $(k, \ell))$ 。

$$\phi_{i j, i^{\prime} j^{\prime}}(\omega) \equiv-\phi_{i^{\prime} j^{\prime}, i j}(\omega) \quad \forall\left(i j, i^{\prime} j^{\prime} \in \mathcal{J}\right) \quad \int_{\Omega} \mathrm{e}^{-\phi_{i j, j^{\prime} j^{\prime}}(\omega)} p_{A(x)}(\omega) \Pi(d \omega) \leq \epsilon_{i j, i^{\prime} j^{\prime}}$$

$$\forall\left((i, j) \in \mathcal{J},\left(i^{\prime}, j^{\prime}\right) \in \mathcal{J},\left(i j, i^{\prime} j^{\prime}\right) \notin \mathcal{C} \alpha, \beta\right): A(X i j) \cap A\left(X_{i^{\prime} j^{\prime}}\right)=\emptyset .$$

## 英国补考|统计推断代写Statistical inference代考|A modification

1. 我们将假设集的“切片” :和 (等效地，和是是 – 彼此接近，即 $\mathcal{C} \alpha \beta^{j} H i j, i \in \mathcal{I} j$
\begin{aligned} &C \alpha, \beta \ &\left.H_{i j} H_{i^{\prime} j} i i^{\prime}\right) \mathcal{C} \alpha, \beta^{j}\left(i j, i^{\prime} j\right) \mathcal{C} \alpha, \beta \ &\left|x_{i}-x_{i^{\prime}}\right| \leq 2 \bar{\alpha} r_{j}+\beta, \bar{\alpha}=\frac{\alpha-1}{2} . \end{aligned}
2. 我们构建了最优检测器，以及它们的风险，对于所有使得。如果对于后一种类型 的对，即 $\phi_{i j, i^{\prime} j} \epsilon_{i j, i^{\prime} j} i, i^{\prime} \in \mathcal{I} j\left(i, i^{\prime}\right) \notin \mathcal{C} \alpha, \beta^{j} \epsilon_{i j, i^{\prime} j}=1 i, i^{\prime}$
3. $A\left(X_{i j}\right) \cap A\left(X_{i^{\prime} j}\right) \neq \emptyset$ ，我们声称是不可接受的并停止。否则我们找到最小 的使得对称矩阵的谱范数与条目 $(\alpha, \beta) K=K_{j} M \times M E^{j K}$
4. $\$ \$$5. E_{ii^{1prime }} \wedge j K}=\ \$$ 不超过。然后我们使用第节的机制来构建基于检吅器的
$$\begin{gathered} \left{\epsilon_{i j, i^{\prime} j}^{K}, \quad i \in \mathcal{I} j, i^{\prime} \in \mathcal{I} j,\left(i, i^{\prime}\right) \notin \mathcal{C} \alpha, \beta^{j} 0, \quad\right. \text { otherwise } \ \bar{\epsilon}=\epsilon /(N+1) 2.5 .2 .3 \mathcal{T} \mathcal{C} \alpha, \beta^{j}{ }^{K j} H_{i j}, i \in \mathcal{I} j \mathcal{C} \alpha, \beta^{j} \bar{\epsilon} \end{gathered}$$

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