Empirical Distribution Function with Confidence Band

The empirical distribution function is the step function giving the proportion of observations at or below each value, shown here with a Dvoretzky–Kiefer–Wolfowitz simultaneous confidence band.

각 값 이하인 관측의 비율을 나타내는 계단 함수인 경험분포함수를 Dvoretzky–Kiefer–Wolfowitz 동시 신뢰대와 함께 추정하는 방법입니다.

I want to show the whole distribution of waiting times with uncertainty, not just the mean. How do I put a confidence band around the cumulative distribution?

평균만이 아니라 대기 시간 분포 전체를 불확실성과 함께 보여 주고 싶어요. 누적분포에 신뢰대를 어떻게 그리나요?

Plot the empirical distribution function and add the Dvoretzky–Kiefer–Wolfowitz band F̂(x) ± ε with ε = √(ln(2/α)/(2n)), which covers the true distribution function at all x simultaneously with probability at least 1 − α. Quantiles can be read off the curve, and reading the band edges gives distribution-free intervals for them.

When to use Empirical Distribution Function with Confidence Band

• You want a distribution-free description of a single variable.
• You need simultaneous uncertainty for the whole cumulative curve.
• You want to compare quantiles across a few groups.

When to use something else

• You need a smooth density estimate; use kernel density estimation instead.
• Data are right-censored; use the Kaplan–Meier estimator instead.

Assumptions and requirements

• Observations are independent and identically distributed.
• No distributional shape is assumed.

What the analysis reports

• ECDF values at observed points with the DKW band and pointwise Wilson intervals
• Band half-width ε
• Selected quantiles with band-based intervals
• Two-sample Kolmogorov–Smirnov test when two groups are given

How to report it

The empirical distribution function of [variable] (n = [value]) was estimated with a 95% Dvoretzky–Kiefer–Wolfowitz simultaneous band (ε = [value]); the median was [value] (band-based interval [[value], [value]]).

Related methods

• Kolmogorov-Smirnov Test
• Kaplan-Meier Estimator
• Distribution Fitting and Selection by Maximum Likelihood

Run Empirical Distribution Function with Confidence Band on Quorant