Mathematical Statistics Lesson of the Day – The Glivenko-Cantelli Theorem

In 2 earlier tutorials that focused on exploratory data analysis in statistics, I introduced

There is actually an elegant theorem that provides a rigorous basis for using empirical CDFs to estimate the true CDF – and this is true for any probability distribution.  It is called the Glivenko-Cantelli theorem, and here is what it states:

Given a sequence of n independent and identically distributed random variables, X_1, X_2, ..., X_n,

P[\lim_{n \to \infty} \sup_{x \epsilon \mathbb{R}} |\hat{F}_n(x) - F_X(x)| = 0] = 1.

In other words, the empirical CDF of X_1, X_2, ..., X_n converges uniformly to the true CDF.

My mathematical statistics professor at the University of Toronto, Keith Knight, told my class that this is often referred to as “The First Theorem of Statistics” or the “The Fundamental Theorem of Statistics”.  I think that this is a rather subjective title – the central limit theorem is likely more useful and important – but Page 261 of John Taylor’s An introduction to measure and probability (Springer, 1997) recognizes this attribution to the Glivenko-Cantelli theorem, too.

2 Responses to Mathematical Statistics Lesson of the Day – The Glivenko-Cantelli Theorem

  1. Justin Rising says:

    There’s actually a stronger result known as the Dvoretzky-Kiefer-Wolfowitz inequality (http://en.wikipedia.org/wiki/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality) which gives a probabilistic bound on how far off the empirical CDF is from the true CDF at any given sample size. It deserves to be better known.

Your thoughtful comments are much appreciated!

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