PPPlot

PPPlot, short for P-P plot, is a graphical method used in statistics to compare probability distributions. It can be used to assess how well a sample agrees with a theoretical distribution or to compare two empirical distributions by examining their cumulative distribution functions (CDFs).

Construction and interpretation:

- When comparing a sample to a theoretical distribution F0, sort the data x(1) ≤ x(2) ≤ ... ≤ x(n). For each i, compute the empirical CDF value i/n and the CDF value F0(x(i)). Plot the points (i/n, F0(x(i))). If the data follow F0, the points should lie near the diagonal y = x. Deviations indicate departures from the reference distribution, such as skewness or heavy tails.

- For two samples, plot the empirical CDF of one sample against the empirical CDF of the other, i.e., F1_hat(x) versus F2_hat(x). Agreement again appears near the diagonal.

Relation to other plots:

- PP plots are related to Q-Q plots but compare probabilities (CDF values) rather than quantiles. They can be more sensitive in certain regions of the distribution, particularly the center, though they may be less informative about tails in some situations.

Uses and limitations:

- PP plots are commonly used for goodness-of-fit assessment and model diagnostic checks, including normality and other parametric families.

- Limitations include sensitivity to sample size and ties in the data; interpretation can be less straightforward for discrete data or when distributions differ in shape only in small regions.

Software implementations exist in various statistical environments, often as part of goodness-of-fit plotting utilities.