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Distribution Map Command

The distribution_map command creates an interactive skewness-kurtosis map. It compares theoretical probability distributions with empirical data series in moment space.

Example

distribution_map(LOGDIFF(SP500))

This example plots the skewness and kurtosis of S&P 500 log returns against common probability distributions.

Multiple Series Example

data = LIST(
    LOGDIFF(SP500),
    LOGDIFF(VIXCLS),
    LOGDIFF(US_ZERO10)
);

distribution_map(data)

This compares equity returns, volatility changes, and Treasury yield changes on the same distribution map.

Syntax

distribution_map()
distribution_map(series)
distribution_map(seriesList)

With no arguments, the command displays the theoretical distribution map. With one or more data series, it also plots each empirical series as a red point. A DataSeriesList is automatically aligned to the same date range before the moments are calculated.

Interpretation

The horizontal axis shows skewness, γ₃. Skewness measures asymmetry. Negative skewness indicates larger downside observations. Positive skewness indicates larger upside observations.

The vertical axis shows kurtosis, γ₄. Kurtosis measures tail thickness. Higher kurtosis indicates more extreme observations than a Normal distribution.

Blue points are reference distributions. Gray curves and shaded regions show distribution families as their parameters change. Red points are the user-supplied data series.

The dashed Pearson lower bound shows the minimum possible kurtosis for a given skewness. Because the chart uses an inverted kurtosis axis, valid distributions lie on or below this curve.

Typical Use Cases

Compare financial return distributions with Normal, Student-t, Gamma, Weibull, and Lognormal families.
Identify heavy-tailed behavior in equity, volatility, interest-rate, or macroeconomic series.
Show whether a series is symmetric, negatively skewed, positively skewed, or fat-tailed.
Teach how probability distributions differ in shape.

Notes

The command uses Pearson kurtosis, where the Normal distribution has kurtosis equal to 3. If excess kurtosis is calculated internally, 3 is added before plotting.

This map is inspired by: Vargo, E., Pasupathy, R., and Leemis, L. M. (2010), Skewness and Kurtosis in Probability Distributions, Journal of Quality Technology, 42(4), 355–377.