The Empirical Distribution Function

Definition (Empirical Distribution Function)

The empirical distribution function is the CDF that puts mass at each data point . Formally,

where is the indicator function.

Theorem

For any fixed value of ,

The DKW Inequality

The following inequality bounds the probability that the random function differs from by more than given constant anywhere on the real line.

Theorem (The Dvoretzky-Kiefer-Wolfowitz (DKW) Inequality)

Let . Then, for any ,

Building CDF Bands

From the DKW inequality, we can construct a confidence bound as follows:

Definition (A Nonparametric Confidence Band for )

Define,

It follows that for any ,

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 7.1.