Introduction

Skewness and kurtosis are vital statistical measures that provide insights into the shape and distribution characteristics of a random variable. In this chapter, we will delve into the estimation and interpretation of skewness and kurtosis, enabling us to understand the symmetry, tail behavior, and concentration of data points in a distribution.

Skewness refers to the degree of asymmetry in a distribution, indicating whether the data is skewed to the left or right. Kurtosis, on the other hand, assesses the concentration of data around the mean and the tails’ thickness. Understanding these measures is crucial for various fields, including finance, where assessing the risk and return distribution of assets is paramount.

Estimating Skewness

To estimate skewness for a sample, we employ the formula:

$$S=\frac{n}{(n-1)(n-2)} \sum_{i=1}^n \frac{\left(x_i-\bar{x}\right)^3}{s^3}$$

Here, $n$ is the sample size, $x_i$ represents individual data points, $\bar{x}$ is the sample mean, and $s$ is the sample standard deviation.

• Positive skewness indicates a longer right tail, while negative skewness suggests a longer left tail.

Estimating Kurtosis

Kurtosis estimation involves the formula:

$$K=\frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^n \frac{\left(x_i-\bar{x}\right)^4}{s^4}-\frac{3(n-1)^2}{(n-2)(n-3)}$$

• Higher kurtosis implies a sharper peak and fatter tails compared to a normal distribution.

• Kurtosis can be compared to the kurtosis of a normal distribution (which is 3 ) to understand the distribution’s shape.

Interpretation of Skewness and Kurtosis

• Positive skewness indicates that the distribution’s tail is longer on the right, implying more extreme values on the positive side.

• Negative skewness suggests a longer left tail, signifying more extreme values on the negative side.

• Kurtosis values above 3 indicate heavy tails and a peaked distribution (leptokurtic), while values below 3 indicate lighter tails and a flatter distribution (platykurtic).

Example: Consider a dataset of stock returns. If the skewness is 0.7, it indicates a slightly positive skew, implying that there are more extreme positive returns. If the kurtosis is 4.2, the distribution has higher kurtosis than a normal distribution, indicating fatter tails and a more pronounced peak.