Dvoretzky–Kiefer–Wolfowitz inequality
In the theory of
with an unspecified multiplicative constant C in front of the exponent on the right-hand side.[1]
In 1990, Pascal Massart proved the inequality with the sharp constant C = 2,[2] confirming a conjecture due to Birnbaum and McCarty.[3] In 2021, Michael Naaman proved the multivariate version of the DKW inequality and generalized Massart's tightness result to the multivariate case, which results in a sharp constant of twice the dimension k of the space in which the observations are found: C = 2k.[4]
The DKW inequality
Given a natural number n, let X1, X2, …, Xn be real-valued
so is the probability that a single random variable is smaller than , and is the fraction of random variables that are smaller than .
The Dvoretzky–Kiefer–Wolfowitz inequality bounds the probability that the
which also implies a two-sided estimate[5]
This strengthens the
with equality if and only if F is continuous.
Multivariate case
In the multivariate case, X1, X2, …, Xn is an i.i.d. sequence of k-dimensional vectors. If Fn is the multivariate empirical cdf, then
for every ε, n, k > 0. The (n + 1) term can be replaced with a 2 for any sufficiently large n.[4]
Kaplan–Meier estimator
The Dvoretzky–Kiefer–Wolfowitz inequality is obtained for the Kaplan–Meier estimator which is a right-censored data analog of the empirical distribution function
for every and for some constant , where is the Kaplan–Meier estimator, and is the censoring distribution function.[7]
Building CDF bands
The Dvoretzky–Kiefer–Wolfowitz inequality is one method for generating CDF-based confidence bounds and producing a confidence band, which is sometimes called the Kolmogorov–Smirnov confidence band. The purpose of this confidence interval is to contain the entire CDF at the specified confidence level, while alternative approaches attempt to only achieve the confidence level on each individual point, which can allow for a tighter bound. The DKW bounds runs parallel to, and is equally above and below, the empirical CDF. The equally spaced confidence interval around the empirical CDF allows for different rates of violations across the support of the distribution. In particular, it is more common for a CDF to be outside of the CDF bound estimated using the DKW inequality near the median of the distribution than near the endpoints of the distribution.
The interval that contains the true CDF, , with probability is often specified as
which is also a special case of the asymptotic procedure for the multivariate case,[4] whereby one uses the following critical value
for the multivariate test; one may replace 2k with k(n + 1) for a test that holds for all n; moreover, the multivariate test described by Naaman can be generalized to account for heterogeneity and dependence.
See also
- Concentration inequality – a summary of bounds on sets of random variables.
References
- MR 0083864
- MR 1062069
- Zbl 0087.34002.
- ^ S2CID 233844405.
- ISBN 9780387749778
- ^
ISBN 0-471-86725-X
- ^ Bitouze, D.; Laurent, B.; Massart, P. (1999), "A Dvoretzky–Kiefer–Wolfowitz type inequality for the Kaplan–Meier estimator", Annales de l'Institut Henri Poincaré B, 35 (6),