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Interpreting the Q-Q Plot with Gumbel Distribution

Q-Q plot for Gumbel distribution

Understanding the Q-Q Plot

The Quantile-Quantile (Q-Q) plot shown above is a diagnostic tool used to evaluate how well a fitted probability distribution—in this case, the Gumbel distribution—matches the observed data. Q-Q plots are particularly useful in extreme value analysis when you want to verify that your data conform to a theoretical distribution.

What Does the Q-Q Plot Show?

  • X-axis (Theoretical Quantiles): These are the expected values from the Gumbel distribution based on your sample size.
  • Y-axis (Ordered Values): These are the actual sorted (empirical) annual maxima from your dataset.
  • Blue Dots: Each point represents a quantile pair—one from the theoretical Gumbel model, and one from the observed dataset.
  • Red Line: This is the reference line where theoretical = empirical. The closer the blue points lie to this line, the better the fit.

How to Interpret the Fit

The alignment of blue dots with the red reference line is a visual indicator of goodness-of-fit. In this plot:

  • Good central fit: In the mid-range (quantiles 25 to 35), the points are nearly on the line. This shows that the Gumbel distribution fits well in the core of the data.
  • Right tail deviation: For higher quantiles (40+), the dots rise above the line. This suggests that the empirical data have heavier right tails than predicted by the Gumbel model.
  • Left tail match: The lower quantiles (below 25) also closely follow the line, meaning the left side of the distribution is well modeled.

Why It Matters

A Q-Q plot like this is useful for:

  • Validating your distribution choice in extreme value theory.
  • Supporting statistical testing (e.g., KS or Shapiro-Wilk) with visual confirmation.
  • Improving confidence in return period estimates for rare events.

Conclusion

This Q-Q plot confirms that the Gumbel distribution provides a fairly strong fit to the central and lower range of annual maximum data, with minor deviations in the upper tail. Such deviations may be acceptable depending on the level of risk you are modeling. Always complement this plot with other diagnostics like the CDF, P-P plot, and goodness-of-fit statistics.

Published by AgriMetSoft