Before fitting any parametric model to daily returns, a desk needs a model-free way to ask "does the normal distribution actually look right here?" — and the answer starts with the empirical CDF, the data's own distribution function, built directly from the order statistics of the previous lesson with no parametric assumption at all. This note defines the ECDF, states the Glivenko–Cantelli theorem that justifies treating it as a consistent stand-in for the true (unknown) CDF, and builds the QQ-plot — the single most useful diagnostic plot in applied statistics — from exactly the quantile machinery of the previous lesson. It closes on the canonical finance application: does a QQ-plot of daily equity returns against the normal distribution actually look straight, or does it reveal the fat tails everyone already suspects?
Learning objectives. By the end of this note you should be able to:
- define the empirical CDF and compute its mean, variance, and jump structure at any fixed point;
- state the Glivenko–Cantelli theorem and explain what "uniform" convergence adds beyond pointwise consistency;
- construct a QQ-plot from sample order statistics and a reference distribution;
- read a QQ-plot's shape to diagnose skewness, heavy tails, and location/scale shifts;
- explain why a QQ-plot of daily equity returns against the normal typically bends away from a straight line in the tails.
In this lesson
- The empirical CDF
- Glivenko–Cantelli: uniform convergence of the ECDF
- QQ-plots
Before you start
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