Simultaneous empirical distribution confidence band ($[L_n(x),U_n(x)]$)

Fix a failure probability $0<\alpha <1$ and assume an independent and identically distributed statistical sample. A finite-sample confidence band for its population cumulative distribution function isHere $\log$ is the natural logarithm. The DKW inequality impliesThis is simultaneous coverage over the whole real line, valid at each fixed sample size.^[ref]

To obtain the band, solve $2e^{-2n{\epsilon }^2}=\alpha$ for the positive $\epsilon$. Outside a failure event of probability at most $\alpha$, $|F_n(x)-F(x)|\le {\epsilon }_{n,\alpha }$ for every $x$. Since a cumulative distribution function already lies in $[0,1]$, clipping the endpoints to this interval preserves the coverage event. The guarantee is at least $1-\alpha$, and need not equal it.

Choose $\alpha =0.1$, giving at least $90\%$ simultaneous coverage under the stated model. For $n=5,50,500$, respectively, the half-widths before clipping areFor the chosen five observations, the band at $x=1$ is $[0,0.94733]$, while at $x=4$ it is approximately $[0.25267,1]$. At $x=3$ the unclipped band would be $[0.05267,1.14733]$, so clipping gives $[0.05267,1]$. The broad uncertainty is informative: five observations give a precise description of themselves, but a coarse bound for an unknown population cumulative distribution function.

A collection of $90\%$ pointwise confidence intervals only promises coverage separately at each preselected threshold; it does not promise $90\%$ coverage of an entire cumulative distribution function. Here the one simultaneous event permits inspecting any threshold after seeing the data. The probability refers to repeated sampling of the entire statistical sample, with $F$ fixed. It does not assign $90\%$ probability to a fixed $F$ after observing this particular band, and it is not a prediction interval for a future observation. The displayed $n=5,50,500$ comparison gives a guarantee for each fixed $n$ separately, not simultaneous validity over all sample sizes or an optional stopping rule. The chosen observations are an illustration; the coverage claim follows from the DKW inequality, not from their appearance or a simulation. Dependence or observations with different sampling laws require another argument.

At fixed $\alpha$, multiplying $n$ by $100$ divides the half-width by $10$. Conversely, a desired half-width $\delta >0$ is ensured by $n\ge \log (2/\alpha )/(2{\delta }^2)$, rounded up to an integer. For $\alpha =0.1$ and $\delta =0.1$, this sufficient sample size is $150$. This makes the confidence band useful for planning how many observations are needed before estimating threshold probabilities to a chosen simultaneous accuracy.