Empirical cumulative distribution functions

Describes the relative frequency for every interval for a given Ordered Data Sequence.

$$F_n(t):=h_n((-\infty ,t])=\frac{1}{n}\sum _{i=1}^n{1}_{(-\infty ,t]}\left(x_i\right)$$

• limit to minus infinity goes to 0 and limit to infinity goes to 1.
• it is weackly monotonically increasing
• it is right continuous
• it is piecewise constant

All of these properties carry over to the Cumulative Distribution Function.

Relative Frequencies for intervals can be calculated: $h_n((a,b])=F_n(b)-F_n(a)$

For n to infinity the ECDF converges to the Cumulative Distribution Function (CDF)

Distribution of Estimator For a CDF $F$ of $X_1$ we have Expectation and Variance of an ECDF $\mathbb{E}\left(F\left(t;X_1,\dots ,X_n\right)\right)=F(t),\mathbb{V}\left(F\left(t;X_1,\dots ,X_n\right)\right)=\frac{F(t)\cdot (1-F(t))}{n}$

Proofs

• Proof Monotonic Increase
• Proof of Right Continuity