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• Statistical Inference 统计推断
• Statistical Computing 统计计算
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统计代写|统计推断代写Statistical inference代考|TESTING MULTIPLE HYPOTHESES

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.
2.5.1.2 The construction
In the situation just described, let $\phi_{i j}$ be the detectors underlying the tests $\mathcal{T}{i j}$; w.l.o.g., we can assume these detectors balanced (see Section 2.3.2.2) with some risks $\epsilon{i j}$ :
$$\left.\begin{array}{ll} \int_{\Omega} \mathrm{e}^{-\phi_{i j}(\omega)} P(d \omega) \leq \epsilon_{i j} & \forall P \in \mathcal{R}{i} \ \int{\Omega} \mathrm{e}^{\phi_{i j}(\omega)} P(d \omega) \leq \epsilon_{i j} & \forall P \in \mathcal{B}{j} \end{array}\right}, 1 \leq i \leq r, 1 \leq j \leq b .$$ Let us assemble the detectors $\phi{i j}$ into a detector for $R, B$ as follows:
$$\phi(\omega)=\max {1 \leq i \leq r} \min {1 \leq j \leq b}\left[\phi_{i j}(\omega)-\alpha_{i j}\right],$$
where the shifts $\alpha_{i j}$ are parameters of the construction.

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.

Risks of an “up to closeness” test. The notion of $\mathcal{C}$-risk of a test was introduced in Section 2.2.4.2, we reproduce it here for the reader’s convenience. Given closeness $\mathcal{C}$ and a test $\mathcal{T}$, we define the $\mathcal{C}$-risk
$$\operatorname{Risk}^{\mathcal{C}}\left(\mathcal{T} \mid H_{1}, \ldots, H_{L}\right)$$
of $\mathcal{T}$ as the smallest $\epsilon \geq 0$ such that
Whenever an observation $\omega$ is drawn from a distribution $P \in \bigcup_{\ell} \mathcal{P}{\ell}$, and $\ell{}$ is such that $P \in \mathcal{P}{\ell{0}}$ (i.e., hypothesis $H_{\ell_{0}}$ is true), the $P$-probability of the event $\ell_{} \notin \mathcal{T}(\omega)$ (“true hypothesis $H_{\ell_{}}$ is not accepted”) or there exists $\ell^{\prime}$ not close to $\ell$ such that $H_{\ell}$ is accepted” is at most $\epsilon$. Equivalently: $\operatorname{Risk}^{\mathcal{C}}\left(\mathcal{T} \mid H_{1}, \ldots, H_{L}\right) \leq \epsilon$ if and only if the following takes place: Whenever an observation $\omega$ is drawn from a distribution $P \in \bigcup_{\ell} \mathcal{P}{\ell}$, and $\ell{}$ is such that $P \in \mathcal{P}{\ell{}}$ (i.e., hypothesis $H_{\ell_{0}}$ is true), the $P$-probability of the event $\ell_{} \in \mathcal{T}(\omega)$ (“the true hypothesis $H_{\ell_{0}}$ is accepted”) and $\ell^{\prime} \in \mathcal{T}(\omega)$ implies that $(\ell, \ell) \in \mathcal{C}$ (“all accepted hypotheses are $\mathcal{C}$-close to the true hypothesis $H_{\ell_{*}} “$ ) is at least $1-\epsilon$.

统计代写|统计推断代写Statistical inference代考|Illustration: Selecting the best among a family of estimates

Let us illustrate our machinery for multiple hypothesis testing by applying it to the situation as follows:
We are given:

• a simple nondegenerate observation scheme $\mathcal{O}=\left(\Omega, \Pi ;\left{p_{\mu}(\cdot): \mu \in\right.\right.$ $\mathcal{M}} ; \mathcal{F})$
• a seminorm $|\cdot|$ on $\mathbf{R}^{n, 11}$
• a convex compact set $X \subset \mathbf{R}^{n}$ along with a collection of $M$ points $x_{i} \in$ $\mathbf{R}^{n}, 1 \leq i \leq M$, and a positive $D$ such that the $|\cdot|$-diameter of the set $X^{+}=X \cup\left{x_{i}: 1 \leq i \leq M\right}$ is at most $D:$
$$\left|x-x^{\prime}\right| \leq D \forall\left(x, x^{\prime} \in X^{+}\right),$$
• an affine mapping $x \mapsto A(x)$ from $\mathbf{R}^{n}$ into the embedding space of $\mathcal{M}$ such that $A(x) \in \mathcal{M}$ for all $x \in X$,
• a tolerance $\epsilon \in(0,1)$.
We observe a $K$-element sample $\omega^{K}=\left(\omega_{1}, \ldots, \omega_{K}\right)$ of observations
$$\omega_{k} \sim p_{A\left(x_{}\right)}, 1 \leq k \leq K,$$ independent across $k$, where $x_{} \in \mathbf{R}^{n}$ is an unknown signal known to belong to $X$. Our “ideal goal” is to use $\omega^{K}$ in order to identify, with probability $\geq 1-\epsilon$, the $|\cdot|$-closest to $x_{*}$ point among the points $x_{1}, \ldots, x_{M}$.
The goal just outlined may be too ambitious, and in the sequel we focus on the relaxed goal as follows:

Given a positive integer $N$ and a “resolution” $\theta>1$, consider the grid
$$\Gamma=\left{r_{j}=D \theta^{-j}, 0 \leq j \leq N\right}$$
and let
$$\rho(x)=\min \left{\rho_{j} \in \Gamma: \rho_{j} \geq \min {1 \leq i \leq M}\left|x-x{i}\right|\right} .$$
Given the design parameters $\alpha \geq 1$ and $\beta \geq 0$, we want to specify a volume of observations $K$ and an inference routine $\omega^{K} \mapsto i_{\alpha, \beta}\left(\omega^{K}\right) \in{1, \ldots, M}$ such that
$$\forall\left(x_{} \in X\right): \operatorname{Prob}\left{\left|x_{}-x_{i_{\alpha, \beta}\left(\omega^{K}\right)}\right|>\alpha \rho\left(x_{*}\right)+\beta\right} \geq 1-\epsilon .$$

统计推断代考

MULTIPLE HYPOTHESES

$R_{i}: P \in \mathcal{R} i$ (止色假设) 和 $B j: P \in \mathcal{B} j$ (蓝色假设)：假讵对于每个 $i \leq r, j \leq b \mathrm{~ 俄 们 有 一 个 简 单 的 基 于 检 则 器 的 测 试 供 㧴}$ 能䏧夫定 $R_{i}$ 对比 $B_{j} \mathrm{~ 涐 们 相 要 的 是 把 这 㓙}$
$$R: P \in \mathcal{R}:=\bigcup_{i=1}^{r} \mathcal{R} i, \quad B: P \in \mathcal{B}:=\bigcup^{j=1^{b} \mathcal{B} j} .$$

$\mathrm{~ L e f t ~ 的 分 陾}$

$\mathrm{~ l e f t ~ 的 份 䘏 答}$

2.5.1.2构造

$\mathrm{~ 到 目 前 为 止 ， 我 伌 已 经 者 䖉 了 基 于 检 则 器 的 䈕 ⿻}$ 2.5.1节)。仞现在涐们打算考虑测试多个 (可能超过 $2 \mathrm{~ 个 ) ~ 假 设 直 至 接 近 ” ； 后 一 个}$

• C包含对角线: $(\ell, \ell) \in \mathcal{C}$ 对于每个 $\ell \leq L \mathrm{~ ( ” 每 个 徦 设 都 阠 自 弟}$
• $\mathcal{C}$ 是对称的: 毎当 $\left(\ell, \ell^{\prime}\right) \in \mathcal{C} \mathrm{~ ， 我 行 ⿰}$
回相一下测试 $\mathcal{T}$ ค定假讵 $H_{1}, \ldots, H_{L}$ 通过观㝗 $\omega \in \Omega \mathrm{~ 是 一 个 过 程 ， 在 圩}$ $H \ell$ 和 $\ell \in \mathcal{T}(\omega)$ ，并拒绝所有其他假设。
$\mathrm{~ 接 近 侏 财 沅}$ 定义 $\mathcal{C}$-风验
$$\operatorname{Risk}^{\mathcal{C}}\left(\mathcal{T} \mid H_{1}, \ldots, H_{L}\right)$$
的 $\mathcal{T}$ 作为最小的 $\epsilon \geq 0$ 这样 受 ) 或存在 $\ell^{\prime}$ 不接近 $\ell$ 这样 $H \ell^{\text {被接受”最冬 }}$ $P \in \bigcup_{\ell} \mathcal{P} \ell_{r}$ ，和 $\ell$ 是这样的 $P \in \mathcal{P} \ell$ (即假设 $H_{\ell_{0}}$ 是真的)， $P \mathrm{~ – ~ 事 朱 的 狔 ल ⿰}$ $(\ell, \ell) \in \mathcal{C}$ (“所有公认的假设都是 $\mathcal{C}$ – 接近真实假设 $H_{\ell+}$ (I) 至”是 $1-\epsilon$.
统计代写|统计推断代写Statistical inference代考|Illustration: Selecting the best among a family of estimates
$\mathrm{~ 让 我 | 门 逦 过 将 其 应 用 于 以 下 情 兄 来 说 明 我 i}$ 戓们得到:
$\mathrm{~ – ~ 一 个 算 哩 的 非 退 化 观 㟯 方 宴 ~ l e f t ~ 的 分 伵}$
• 半规范 $|\cdot|$ 上 $\mathbf{R}^{n, 11}$
• 凸䒨集 $X \subset \mathbf{R}^{n}$ 连同一䒺列 $M$ 祸分 $x_{i} \in \mathbf{R}^{n}, 1 \leq i \leq M$, 和一个正 $D$ 使得 $\mid \boldsymbol{-}$
$\mathrm{~ M e f t ~ 的 分 伃 符 谋}$
最侈是 $D$
$$\left|x-x^{\prime}\right| \leq D \forall\left(x, x^{\prime} \in X^{+}\right),$$
• 仿买映射 $x \mapsto A(x) 从 \mathbf{R}^{n}$ 进入嵌入空间 $\mathcal{M}$ 这样 $A(x) \in \mathcal{M}$ 对所有人 $x \in X$,
• 宽容 $\epsilon \in(0,1)$.
我们观窖到一个 $K \mathrm{~ – 元 龺}$
$$\omega_{k} \sim p_{A(z)}, 1 \leq k \leq K,$$ 中点 $x_{1}, \ldots, x_{M}$
$\mathrm{~ 刏 才 勾 勒 的 目 标 可 能 扎 于 宏 大 ， 后 経 㧴 们 洚 重 点 放 在 案}$ 给定一个正蝗数 $N$ 和一个”决议 $\theta>1 \mathrm{~ ， 考 虑 囘}$
$\mathrm{~ l e f t ~ 的 分 哃}$
然后让
$\mathrm{~ M e f t ~ 的 攽 陗 符 㘴 先 或 ⿰}$

有限元方法代写

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MATLAB代写

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