Introduction

In the realm of statistical analysis, moments play a pivotal role in understanding the characteristics of a distribution. We’ve explored how skewness and kurtosis provide insights into the asymmetry and peakedness of a distribution, respectively. In this chapter, we’ll delve into the concepts of coskewness and cokurtosis, shedding light on how these higher-order moments relate to skewness and kurtosis, respectively. By exploring these relationships, we deepen our understanding of the distribution’s behavior and uncover nuances that mere skewness and kurtosis might overlook.

Coskewness

Coskewness takes us beyond the realm of a single variable’s skewness. It measures the degree to which two variables exhibit similar or different skewness patterns in relation to a third variable. Essentially, it captures the tendency of two variables to deviate from their means in the same direction. In mathematical terms, for three variables $X$, $Y$, and $Z$:

$$Coskewness(X, Y, Z) = E[(X - μ_X)(Y - μ_Y)(Z - μ_Z)]$$

Where:

• $E$ is the expectation operator
• $μ_X$, $μ_Y$, and $μ_Z$ are the means of variables $X$, $Y$, and $Z$

Cokurtosis

Similarly, cokurtosis extends the idea of kurtosis by analyzing how the tails of two variables behave relative to a third variable. It indicates the degree to which the tails of two variables exhibit similar or different behaviors. In essence, cokurtosis measures the fourth-order moment of the joint distribution of three variables. For variables $X$, $Y$, and $Z$:

$$Cokurtosis(X, Y, Z)=E[(X-\mu_X)^4(Y-\mu_Y)(Z-\mu_Z)]$$

Where:

• $E$ is the expectation operator
• $\mu_X$, $\mu_Y$, and $\mu_Z$ are the means of variables $X$, $Y$, and $Z$

Understanding the Relationship

The relationship between coskewness, cokurtosis, skewness, and kurtosis is intertwined. Coskewness and skewness both relate to asymmetry but in different ways. Coskewness assesses how two variables respond to changes in a third variable, while skewness provides insight into the distribution’s asymmetry.

On the other hand, cokurtosis and kurtosis both deal with the distribution’s tails. Cokurtosis reveals whether two variables exhibit similar or different tail behaviors compared to a third variable. Meanwhile, kurtosis gauges the distribution’s tail thickness.