The ties that bind: A moment-ratio diagram for probability distributions

A moment-ratio diagram allows a modeller to see several probability distributions at one time. The figure below shows the skewness versus the kurtosis for several probability distributions, where the skewness and kurtosis are defined by

$\gamma _{3}=E\left [ \left ( \frac{X-\mu_{X}}{\sigma_{X}} \right ) ^{3}\right ];\; \; \gamma _{4}=E\left [ \left ( \frac{X-\mu_{X}}{\sigma_{X}} \right ) ^{4}\right ]$

and μ_X and σ_X are the mean and the standard deviation of some implied univariate random variable X.

These moment-ratio diagrams are useful for:

quantifying the proximity between various univariate distributions based on their third and fourth moments;
illustrating the versatility of a particular distribution based on the range of values that the various moments can assume (the two-parameter beta distribution which covers significant areas in the figure below is an example of this versatility);
creating a short list of potential probability models based on the proximity of the point associated with a data set to a particular probability model; and
clarifying the limiting relationships between various well-known distribution families (the Student’s t distribution approaching the standard normal distribution as the degrees of freedom increases is apparent in the figure).

A moment-ratio diagram can also indicate which distributions play a central role in probability and statistics. The normal distribution at the point (0, 3) and the exponential distribution at the point (2, 9) clearly play a central role in the figure.

Further details concerning moment-ratio diagrams is given in Vargo, Pasupathy, and Leemis.¹

FIGURE A moment-ratio diagram of skewness versus kurtosis for several probability distributions.

About the author

Lawrence M. Leemis is a professor in the Department of Mathematics at the College of William & Mary in Williamsburg, Virginia. Raghu Pasupathy is an associate professor in the Department of Statistics at Purdue University, Indiana.

Note

This article is an online companion piece to “Notebook: The ties that bind”, by Lawrence M. Leemis and Raghu Pasupathy, published in the August 2019 print edition.

Reference

Vargo, E., Pasupathy, R. and Leemis, L. (2010) Moment-Ratio Diagrams for Univariate Distributions, Journal of Quality Technology, 42(3), 276-286.

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