Empirical cumulative distribution functions

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

Fn(t):=hn((−∞,t])=1n∑i=1n1(−∞,t](xi)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:

hn((a,b])=Fn(b)−Fn(a)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 FF of X1X_1 we have Expectation and Variance of an ECDF

E(F(t;X1,…,Xn))=F(t),V(F(t;X1,…,Xn))=F(t)⋅(1−F(t))n\mathbb{E}\left(F\left(t ; X_1, \ldots, X_n\right)\right)=F(t), \quad \mathbb{V}\left(F\left(t ; X_1, \ldots, X_n\right)\right)=\frac{F(t) \cdot(1-F(t))}{n}

Proofs