## 统计代写|假设检验代写hypothesis testing代考|R Functions skerd, kerden, kdplot, rdplot, akerd, and splot

It is noted that $\mathrm{R}$ has a built-in function called density that computes a kernel density estimate based on various choices for $K$. (This function also contains various options not covered here.) By default, $K$ is taken to be the standard normal density. Here, the R function
skerd(x,op=T,kernel=”gaussian”)
is supplied in the event there is a desire to plot the data based on this collection of estimators. When $o p=T$, the function uses the default density estimator employed by $R$; otherwise it uses the method recommended by Venables and Ripley (2002, p. 127). (When using R, the default density estimator differs from the one used by S-PLUS, but with op=F, R and S-PLUS use the same method.) To use the Epanechnikov kernel, set the argument kernel=epanechnikov.
The function
$$\operatorname{kerden}(\mathrm{x}, \mathrm{q}=0.5, \mathrm{xval}=0) \text {, }$$ written for this book, computes the kernel density estimate of $f\left(x_q\right)$ for the data stored in the $\mathrm{R}$ vector $x$ using the Rosenblatt shifted histogram method, described in Section 3.2.2. (Again, see Section $1.7$ on how to obtain the functions written for this book.) If unspecified, q defaults to $0.5$. The argument $x v a l$ is ignored unless $\mathrm{q}=0$, in which case the function estimates $f$ when $x$ is equal to value specified by the argument $x v a l$. The function
$$\text { kdplot }(x, r v a l=15)$$
plots the estimate of $f(x)$ based on the function kerden, where the argument rval indicates how many quantiles will be used. The default value, 15 , means that $f(x)$ is estimated for 15 quantiles evenly spaced between $0.01$ and $0.99$, and then the function plots the estimates to form an estimate of $f(x)$.

## 统计代写|假设检验代写hypothesis testing代考|The Sample Trimmed Mean

As already indicated, the standard error of the sample mean can be relatively large when sampling from a heavy-tailed distribution, and the sample mean estimates a nonrobust measure of location, $\mu$. The sample trimmed mean addresses these problems.

The sample trimmed mean, which estimates the population trimmed $\mu_t$ (described in Section 2.2.3), is computed as follows. Let $X_1, \ldots, X_n$ be a random sample and let $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$ be the observations written in ascending order. The value $X_{(i)}$ is called the $i$ th order statistic. Suppose the desired amount of trimming has been chosen to be $\gamma, 0 \leq \gamma<0.5$. Let $g=[\gamma n]$, where $[\gamma n]$ is the value of $\gamma n$ rounded down to the nearest integer. For example, $[10.9]=10$. The sample trimmed mean is computed by removing the $g$ largest and $g$ smallest observations and averaging the values that remain. In symbols, the sample trimmed mean is
$$\bar{X}t=\frac{X{(g+1)}+\cdots+X_{(n-g)}}{n-2 g} .$$
In essence, the empirical distribution is trimmed in a manner consistent with how the probability density function was trimmed when defining $\mu_t$. As indicated in Chapter 2 , two-sided trimming is assumed unless stated otherwise.
The definition of the sample trimmed mean given by Eq. (3.1) is the one most commonly used. However, for completeness, it is noted that the term trimmed mean sometimes refers to a slightly different estimator (e.g., Reed, 1998; cf. Hogg, 1974), namely,
$$\frac{1}{n(1-2 \gamma)}\left(\sum_{i=g+1}^{n-g} X_{(i)}+(g-\gamma n)\left(X_{(g)}+X_{(n-g+1)}\right)\right. \text {. }$$
Also see Patel, Mudholkar, and Fernando (1988) as well as Kim (1992a). Here, however, the definition given by (3.1) is used exclusively.

# 假设检验代考

## 统计代写|假设检验代写hypothesis testing代考|R Functions skerd, kerden, kdplot, rdplot, akerd, and splot

skerd(x,op=T,kernel=”gaussian”)

$$\operatorname{kerden}(mathrm{x}, mathrm{q}=0.5,mathrm{xval}=0) text {, } 为本书编写的$$计算存储在$mathrm{R}$向量$x$中的数据的$f\left(x_q\right)的核密度估计，使用3.2.2节中描述的Rosenblatt移动直方图方法。如果没有指定，q默认为$0.5$。参数$x v a l$被忽略，除非$mathrm{q}=0$，在这种情况下，当$x$等于参数$x v a l$所指定的值时，函数估计$f$。该函数 $$\纹理 { kdplot }(x, r v a l=15)$$ 绘制基于函数kerden的$f(x)$的估计值，其中参数rval表示将使用多少个量纲。默认值为15，意味着$f(x)$被估计为15个均匀分布在$0.01$和$0.99$之间的量值，然后该函数绘制估计值，形成$f(x)$的估计值。 ## 统计代写|假设检验代写hypothesis testing代考|The Sample Trimmed Mean 如前所述，当从重尾分布中取样时，样本平均数的标准误差可能比较大，而且样本平均数估计的是一个非稳健的位置测量值$\mu$。样本修剪平均数可以解决这些问题。 样本修剪平均数估计了人口修剪的$mu_t$（在第2.2.3节中描述），其计算方法如下。让$X_1, \ldots, X_n$为随机样本，让$X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$为按升序排列的观察值。值$X_{(i)}$被称为第i$次统计量。假设所需的修剪量被选为$\gamma, 0\leq \gamma<0.5$。让$g=[\gamma n]$，其中$[\gamma n]$是$\gamma n$的值，四舍五入到最近的整数。例如，$[10.9]=10$。样本修剪平均数的计算方法是去除$g$最大和$g$最小的观测值，并对剩下的值进行平均。在符号中，样本修剪后的平均值为
$$\bar{X}t=frac{X{(g+1)}+\cdots+X_{(n-g)}}{n-2 g}。$$

$$\frac{1}{n(1-2 \gamma)}\left(\sum_{i=g+1}^{n-g} X_{(i)}+(g-\gamma n)\left(X_{(g)}+X_{(n-g+1)}\right)\right. 文字{. }$$

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