# 统计代写|统计推断代写Statistical inference代考|STAT 3023

#### 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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## 统计代写|统计推断代写Statistical inference代考|Situation and goal

Let $\Omega$ be an observation space, and assume we are given two finite collections of families of probability distributions on $\Omega$ : families of red distributions $\mathcal{R}{i}, 1 \leq i \leq r$, and families of blue distributions $\mathcal{B}{j}, 1 \leq j \leq b$. These families give rise to $r$ red and $b$ blue hypotheses on the distribution $P$ of an observation $\omega \in \Omega$, specifically,
$R_{i}: P \in \mathcal{R}{i}$ (red hypotheses) and $B{j}: P \in \mathcal{B}{j}$ (blue hypotheses). Assume that for every $i \leq r, j \leq b$ we have at our disposal a simple detector-based test $\mathcal{T}{i j}$ capable of deciding on $R_{i}$ vs. $B_{j}$. What we want is to assemble these tests into a test $\mathcal{T}$ deciding on the union $R$ of red hypotheses vs. the union $B$ of blue ones:
$$R: P \in \mathcal{R}:=\bigcup_{i=1}^{r} \mathcal{R}{i}, \quad B: P \in \mathcal{B}:=\bigcup{j=1}^{b} \mathcal{B}{j} .$$ Here $P$, as always, stands for the probability distribution of observation $\omega \in \Omega$. Our motivation primarily stems from the case where $R{i}$ and $B_{j}$ are convex hypotheses in a simple o.s. (2.72):
$$\mathcal{R}{i}=\left{p{\mu}: \mu \in M_{i}\right}, \mathcal{B}{j}=\left{p{\mu}: \mu \in N_{j}\right},$$
where $M_{i}$ and $N_{j}$ are convex compact subsets of $\mathcal{M}$. In this case we indeed know how to build near-optimal tests deciding on $R_{i}$ vs. $B_{j}$, and the question we have posed becomes, how do we assemble these tests into a test deciding on $R$ vs. $B$, with
$$\begin{array}{ll} R: P \in \mathcal{R}=\left{p_{\mu}: \mu \in X\right}, & X=\bigcup_{i} M_{i} \ B: P \in \mathcal{B}=\left{p_{\mu}: \mu \in Y\right}, & Y=\bigcup_{j} N_{j} ? \end{array}$$
While the structure of $R, B$ is similar to that of $R_{i}, B_{j}$, there is a significant difference: the sets $X, Y$ are, in general, nonconvex, and therefore the techniques we have developed fail to address testing $R$ vs. $B$ directly.

## 统计代写|统计推断代写Statistical inference代考|Testing multiple hypotheses “up to closeness”

So far, we have considered detector-based simple tests deciding on pairs of hypotheses, specifically, convex hypotheses in simple o.s.’s (Section 2.4.4) and unions of convex hypotheses (Section 2.5.1). ${ }^{10}$ Now we intend to consider testing of multiple (perhaps more than 2) hypotheses “up to closeness”; the latter notion was introduced in Section 2.2.4.2.

Let $\Omega$ be an observation space, and let a collection $\mathcal{P}{1}, \ldots, \mathcal{P}{L}$ of families of probability distributions on $\Omega$ be given. As always, families $\mathcal{P}{\ell}$ give rise to hypotheses $$H{\ell}: P \in \mathcal{P}{\ell}$$ on the distribution $P$ of observation $\omega \in \Omega$. Assume also that we are given a closeness relation $\mathcal{C}$ on ${1, \ldots, L}$. Recall that, formally, a closeness relation is some set of pairs of indices $\left(\ell, \ell^{\prime}\right) \in{1, \ldots, L}$; we interpret the inclusion $\left(\ell, \ell^{\prime}\right) \in \mathcal{C}$ as the fact that hypothesis $H{\ell}$ “is close” to hypothesis $H_{\ell}$. When $\left(\ell, \ell^{\prime}\right) \in \mathcal{C}$, we say that $\ell^{\prime}$ is close (or $\mathcal{C}$-close) to $\ell$. We always assume that

• $\mathcal{C}$ contains the diagonal: $(\ell, \ell) \in \mathcal{C}$ for every $\ell \leq L$ (“each hypothesis is close to itself”), and
• $\mathcal{C}$ is symmetric: whenever $\left(\ell, \ell^{\prime}\right) \in \mathcal{C}$, we have also $\left(\ell^{\prime}, \ell\right) \in \mathcal{C}$ (“if the $\ell$-th hypothesis is close to the $\ell^{\prime}$-th one, then the $\ell^{\prime}$-th hypothesis is close to the $\ell$-th one”).

Recall that a test $\mathcal{T}$ deciding on the hypotheses $H_{1}, \ldots, H_{L}$ via observation $\omega \in \Omega$ is a procedure which, given on input $\omega \in \Omega$, builds some set $\mathcal{T}(\omega) \subset{1, \ldots, L}$, accepts all hypotheses $H_{\ell}$ with $\ell \in \mathcal{T}(\omega)$, and rejects all other hypotheses.

# 统计推断代考

## 统计代写|统计推断代写Statistical inference代考|Situation and goal

$\mathcal{R} i, 1 \leq i \leq r$, 和蓝色分布族 $B j, 1 \leq j \leq b$. 这些家庭引起 $r$ 红色和 $b$ 关于分布的蓝色假 设 $P$ 观賩的 $\omega \in \Omega$ ，具体来说，
$R_{i}: P \in \mathcal{R} i$ (红色假设) 和 $B j: P \in \mathcal{B} j$ (蓝色假设)。假设对于每个 $i \leq r, j \leq b$ 我 们有一个简单的基于检则器的测试供我们使用 $\mathcal{T} i j$ 能够决定 $R_{i}$ 对比 $B_{j}$. 我们想要的是把这 些测试组装成一个测试 $\mathcal{T}$ 决定工会 $R$ 红色假设与联合 $B$ 蓝色的:
$$R: P \in \mathcal{R}:=\bigcup_{i=1}^{r} \mathcal{R} i, \quad B: P \in \mathcal{B}:=\bigcup j=1^{b} \mathcal{B} j .$$

## 统计代写|统计推断代写Statistical inference代考|Testing multiple hypotheses “up to closeness”

$$H \ell: P \in \mathcal{P} \ell$$

• $\mathcal{C}$ 包含对角线: $(\ell, \ell) \in \mathcal{C}$ 对于每个 $\ell \leq L$ (“每个假设都与自身相近”) ，以及
• $\mathcal{C}$ 是对称的: 每当 $\left(\ell, \ell^{\prime}\right) \in \mathcal{C}$ ，我们也有 $\left(\ell^{\prime}, \ell\right) \in \mathcal{C}$ (“如果 $\ell$-th 假设接近 $\ell^{\prime}$-第一 个，然后是 $\ell^{\prime}$-th 假设接近 $\ell$-第一个”)。
回想一下测试 $\mathcal{T}$ 决定假设 $H_{1}, \ldots, H_{L}$ 通过观崇 $\omega \in \Omega$ 是一个过程，在输入时给出 $\omega \in \Omega$ ，建立一些集合 $\mathcal{T}(\omega) \subset 1, \ldots, L$ ，接受所有假设 $H_{\ell}$ 和 $\ell \in \mathcal{T}(\omega)$ ，并拒绝所有其他假设。

## 有限元方法代写

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