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# 统计代写|假设检验作业代写Hypothesis testing代考|STAT311 Scale-Equivariant M-Measures of Location

1778年：皮埃尔-拉普拉斯比较了欧洲多个城市的男孩和女孩的出生率。他说 “很自然地得出结论，这些可能性几乎处于相同的比例”。因此，拉普拉斯的无效假设是，鉴于 “传统智慧”，男孩和女孩的出生率应该是相等的 。

1900: 卡尔-皮尔逊开发了卡方检验，以确定 “给定形式的频率曲线是否能有效地描述从特定人群中抽取的样本”。因此，无效假设是，一个群体是由理论预测的某种分布来描述的。他以韦尔登掷骰子数据中5和6的数量为例 。

1904: 卡尔-皮尔逊提出了 “或然性 “的概念，以确定结果是否独立于某个特定的分类因素。这里的无效假设是默认两件事情是不相关的（例如，疤痕的形成和天花的死亡率）。[16] 这种情况下的无效假设不再是理论或传统智慧的预测，而是导致费雪和其他人否定使用 “反概率 “的冷漠原则。

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## 统计代写| 假设检验作业代写Hypothesis testing代考|Scale-Equivariant M-Measures of Location

M-measures of location can be made scale equivariant by incorporating a measure of scale in the general approach described in Section 2.2.4. That is, rather than determine $\mu_m$ with Eq. (2.10), use
$$E\left[\Psi\left(\frac{X-\mu_m}{\tau}\right)\right]=0,$$
where $\tau$ is some appropriate measure of scale.
When considering which measure of scale should be used in Eq. (2.15), it helps to notice that $\tau$ plays a role in determining whether a value for $X$ is unusually large or small. To illustrate this, consider Huber’s $\Psi$, which, in the present context, is given by
$$\Psi\left(\frac{x-\mu_m}{\tau}\right)= \begin{cases}-K, & \text { if }\left(x-\mu_m\right) / \tau<-K \\ \frac{x-\mu_m}{\tau}, & \text { if }-K \leq\left(x-\mu_m\right) / \tau \leq K \\ K, & \text { if }\left(x-\mu_m\right) / \tau>K\end{cases}$$
Then according to $\Psi$, the distance between $x$ and $\mu_m,\left|x-\mu_m\right|$, is not unusually large or small if $-K \leq\left(x-\mu_m\right) / \tau \leq K$. In this case, the same $\Psi$ used to define the population mean, $\mu$, is being used. If $x-\mu_m>K \tau, \Psi$ considers the distance to be relatively large, and the influence of $x$ on $\mu_m$ is reduced. Similarly, if $x-\mu_m<-K \tau, x$ is considered to be unusually far from $\mu_m$.

## 统计代写|假设检验作业代写HYPOTHESIS TESTING代考|Winsorized Expected Values

One final tool is introduced that has practical value in various situations: Winsorized expected values. What will be needed is a generalization of $E(X)$ that maintains standard properties of expected values.

Let $g(X)$ be any function of the continuous random variable $X$. When working with a single random variable, the $\gamma$-Winsorized expected value of $g(X)$ is defined to be
$$E_w[g(X)]=\int_{x_\gamma}^{x_{1-\gamma}} g(x) d F(x)+\gamma\left[g\left(x_\gamma\right)+g\left(x_{1-\gamma}\right)\right] .$$
That is, the expected value of $g(X)$ is defined in the usual way, only with respect to the Winsorized distribution corresponding to $F$. However, a generalization of $E_w$ is needed that provides Winsorized expected values of linear combinations of random variables.

Let $X$ and $Y$ be any two continuous random variables with joint distribution $F$ and probability density function $f(x, y)$. What is needed is an analog of Winsorization for any bivariate distribution. Note that any point $(x, y)$ falls in one of nine regions shown in Figure 2.3, where the corners of the rectangle are determined by the $\gamma$ and $1-\gamma$ quantiles of $X$ and $Y$. That is, the rectangle is given by the four points $\left(x_\gamma, y_\gamma\right),\left(x_\gamma, y_{1-\gamma}\right),\left(x_{1-\gamma}, y_\gamma\right)$, and $\left(x_{1-\gamma}, y_{1-\gamma}\right)$. Winsorization of any bivariate distribution consists of pulling in any point outside the rectangle formed by these four points, as indicated by the arrows in Figure 2.3. For any point inside this rectangle, the Winsorized distribution has probability density function $f(x, y)$. The corners of the rectangle become discrete distributions, even when working with continuous random variables. For example, the point $\left(x_\gamma, y_\gamma\right)$ has probability $P\left(X \leq x_\gamma, Y \leq y_\gamma\right)$. Similarly, the point $\left(x_\gamma, y_{1-\gamma}\right)$ has probability equal to the probability that $X \leq x_\gamma$ and $Y \geq y_{1-\gamma}$, simultaneously. However, the sides of the rectangle, excluding the four corners, have a continuous distribution when $X$ and $Y$ are continuous.

## 统计代写|假设检验作业代写HYPOTHESIS TESTING代考|SCALE-EQUIVARIANT M-MEASURES OF LOCATION

$$E\left[\Psi\left(\frac{X-\mu_m}{\tau}\right)\right]=0$$

$$\Psi\left(\frac{x-\mu_m}{\tau}\right)= \begin{cases}-K, & \text { if }\left(x-\mu_m\right) / \tau<-K \\ \frac{x-\mu_m}{\top}, & \text { if }-K \leq\left(x-\mu_m\right) / \tau \leq K \\ K, & \text { if }\left(x-\mu_m\right) / \tau>K\end{cases}$$

## 统计代写|假设检验作业代写HYPOTHESIS TESTING代 考|WINSORIZED EXPECTED VALUES

$$E_w[g(X)]=\int_{x_\gamma}^{x_{1-\gamma}} g(x) d F(x)+\gamma\left[g\left(x_\gamma\right)+g\left(x_{1-\gamma}\right)\right]$$

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