### 统计代写|金融统计代写Mathematics with Statistics for Finance代考|SKEWNESS

statistics-lab™ 为您的留学生涯保驾护航 在代写金融统计Mathematics with Statistics for Finance G1GH方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写金融统计Mathematics with Statistics for Finance G1GH方面经验极为丰富，各种代写金融统计Mathematics with Statistics for Finance G1GH相关的作业也就用不着说。

• 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

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|SKEWNESS

The second central moment, variance, tells us how spread-out a random variable is around the mean. The third central moment tells us how symmetrical

the distribution is around the mean. Rather than working with the third central moment directly, by convention we first standardize the statistic. This standardized third central moment is known as skewness:
$$\text { Skewness }=\frac{E\left[(X-\mu)^{3}\right]}{\sigma^{3}}$$
where $\sigma$ is the standard deviation of $X$.
By standardizing the central moment, it is much easier to compare two random variables. Multiplying a random variable by a constant will not change the skewness.

A random variable that is symmetrical about its mean will have zero skewness. If the skewness of the random variable is positive, we say that the random variable exhibits positive skew. Figures $3.1$ and $3.2$ show examples of positive and negative skewness.

Skewness is a very important concept in risk management. If the distributions of returns of two investments are the same in all respects, with the same mean and standard deviation but different skews, then the investment with more negative skew is generally considered to be more risky. Historical data suggest that many financial assets exhibit negative skew.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Negative Skew

As with variance, the equation for skewness differs depending on whether we are calculating the population skewness or the sample skewness. For the population statistic, the skewness of a random variable $X$, based on $n$ observations, $x_{1}, x_{2}, \ldots, x_{n}$, can be calculated as:
$$\hat{s}=\sum_{i=1}^{n}\left(\frac{x_{i}-\mu}{\sigma}\right)^{3}$$
where $\mu$ is the population mean and $\sigma$ is the population standard deviation. Similar to our calculation of sample variance, if we are calculating the sample skewness, there is going to be an overlap with the calculation of the sample mean and sample standard deviation. We need to correct for that. The sample skewness can be calculated as:
$$\tilde{s}=\frac{n}{(n-1)(n-2)} \sum_{i=1}^{n}\left(\frac{x_{i}-\hat{\mu}}{\hat{\sigma}}\right)^{3}$$

Based on Equation $3.20$ for variance, it is tempting to guess that the formula for the third central moment can be written simply in terms of $E\left[X^{3}\right]$ and $\mu$. Be careful, as the two sides of this equation are not equal:
$$E\left[(X-\mu)^{k}\right] \neq E\left[X^{3}\right]-\mu^{3}$$
The correct equation is:
$$E\left[(X-\mu)^{3}\right]=E\left[X^{3}\right]-3 \mu \sigma^{2}-\mu^{3}$$

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|KURTOSIS

The fourth central moment is similar to the second central moment, in that it tells us how spread-out a random variable is, but it puts more weight on extreme points. As with skewness, rather than working with the central moment directly, we typically work with a standardized statistic. This standardized fourth central moment is known as the kurtosis. For a random variable $X$, we can define the kurtosis as $K$, where:
$$K=\frac{E\left[(X-\mu)^{4}\right]}{\sigma^{4}}$$
where $\sigma$ is the standard deviation of $X$, and $\mu$ is its mean.
By standardizing the central moment, it is much easier to compare two random variables. As with skewness, multiplying a random variable by a constant will not change the kurtosis.

The following two populations have the same mean, variance, and skewness. The second population has a higher kurtosis.
Population 1: ${-17,-17,17,17}$
Population 2: ${-23,-7,7,23}$
Notice, to balance out the variance, when we moved the outer two points out six units, we had to move the inner two points in 10 units. Because the random variable with higher kurtosis has points further from the mean, we often refer to distribution with high kurtosis as fat-tailed. Figures $3.3$ and $3.4$ show examples of continuous distributions with high and low kurtosis.
Like skewness, kurtosis is an important concept in risk management. Many financial assets exhibit high levels of kurtosis. If the distribution of

returns of two assets have the same mean, variance, and skewness, but different kurtosis, then the distribution with the higher kurtosis will tend to have more extreme points, and be considered more risky.

As with variance and skewness, the equation for kurtosis differs depending on whether we are calculating the population kurtosis or the sample kurtosis. For the population statistic, the kurtosis of a random variable $X$ can be calculated as:
$$\hat{K}=\sum_{i=1}^{n}\left(\frac{x_{i}-\mu}{\sigma}\right)^{4}$$
where $\mu$ is the population mean and $\sigma$ is the population standard deviation. Similar to our calculation of sample variance, if we are calculating the sample kurtosis, there is going to be an overlap with the calculation of the sample mean and sample standard deviation. We need to correct for that. The sample kurtosis can be calculated as:
$$\tilde{K}=\frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^{n}\left(\frac{x_{i}-\hat{\mu}}{\hat{\sigma}}\right)^{4}$$
In the next chapter we will study the normal distribution, which has a kurtosis of 3 . Because normal distributions are so common, many people refer to “excess kurtosis,” which is simply the kurtosis minus 3 .
$$K_{\text {excess }}=K-3$$
In this way, the normal distribution has an excess kurtosis of 0 . Distributions with positive excess kurtosis are termed leptokurtotic. Distributions with negative excess kurtosis are termed platykurtotic. Be careful; by default, many applications calculate excess kurtosis.

When we are also estimating the mean and variance, calculating the sample excess kurtosis is somewhat more complicated than just subtracting 3. The correct formula is:
$$\tilde{K}_{\text {excess }}=\tilde{K}-3 \frac{(n-1)^{2}}{(n-2)(n-3)}$$
where $\tilde{K}$ is the sample kurtosis from Equation 3.46. As $n$ increases, the last term on the right-hand side converges to 3 .

偏度 =和[(X−μ)3]σ3

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Negative Skew

s^=∑一世=1n(X一世−μσ)3

s~=n(n−1)(n−2)∑一世=1n(X一世−μ^σ^)3

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|KURTOSIS

ķ=和[(X−μ)4]σ4

ķ^=∑一世=1n(X一世−μσ)4

ķ~=n(n+1)(n−1)(n−2)(n−3)∑一世=1n(X一世−μ^σ^)4

ķ过量的 =ķ−3

ķ~过量的 =ķ~−3(n−1)2(n−2)(n−3)

## 广义线性模型代考

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## MATLAB代写

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