# KolmogorovDist

##### Generic function for the computation of the Kolmogorov distance of two distributions

Generic function for the computation of the Kolmogorov distance \(d_\kappa\) of two distributions \(P\) and \(Q\) where the distributions are defined on a finite-dimensional Euclidean space \((\R^m,{\cal B}^m)\) with \( {\cal B}^m \) the Borel-\(\sigma\)-algebra on \(R^m\). The Kolmogorov distance is defined as $$d_\kappa(P,Q)=\sup\{|P(\{y\in\R^m\,|\,y\le x\})-Q(\{y\in\R^m\,|\,y\le x\})| | x\in\R^m\}$$ where \(\le\) is coordinatewise on \(\R^m\).

- Keywords
- distribution

##### Usage

```
KolmogorovDist(e1, e2, ...)
# S4 method for AbscontDistribution,AbscontDistribution
KolmogorovDist(e1,e2)
# S4 method for AbscontDistribution,DiscreteDistribution
KolmogorovDist(e1,e2)
# S4 method for DiscreteDistribution,AbscontDistribution
KolmogorovDist(e1,e2)
# S4 method for DiscreteDistribution,DiscreteDistribution
KolmogorovDist(e1,e2)
# S4 method for numeric,UnivariateDistribution
KolmogorovDist(e1, e2)
# S4 method for UnivariateDistribution,numeric
KolmogorovDist(e1, e2)
# S4 method for AcDcLcDistribution,AcDcLcDistribution
KolmogorovDist(e1, e2)
```

##### Arguments

- e1
object of class

`"Distribution"`

or class`"numeric"`

- e2
object of class

`"Distribution"`

or class`"numeric"`

- …
further arguments to be used in particular methods (not in package distrEx)

##### Value

Kolmogorov distance of `e1`

and `e2`

##### Methods

- e1 = "AbscontDistribution", e2 = "AbscontDistribution":
Kolmogorov distance of two absolutely continuous univariate distributions which is computed using a union of a (pseudo-)random and a deterministic grid.

- e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":
Kolmogorov distance of two discrete univariate distributions. The distance is attained at some point of the union of the supports of

`e1`

and`e2`

.- e1 = "AbscontDistribution", e2 = "DiscreteDistribution":
Kolmogorov distance of absolutely continuous and discrete univariate distributions. It is computed using a union of a (pseudo-)random and a deterministic grid in combination with the support of

`e2`

.- e1 = "DiscreteDistribution", e2 = "AbscontDistribution":
Kolmogorov distance of discrete and absolutely continuous univariate distributions. It is computed using a union of a (pseudo-)random and a deterministic grid in combination with the support of

`e1`

.- e1 = "numeric", e2 = "UnivariateDistribution":
Kolmogorov distance between (empirical) data and a univariate distribution. The computation is based on

`ks.test`

.- e1 = "UnivariateDistribution", e2 = "numeric":
Kolmogorov distance between (empirical) data and a univariate distribution. The computation is based on

`ks.test`

.- e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":
Kolmogorov distance of mixed discrete and absolutely continuous univariate distributions. It is computed using a union of the discrete part, a (pseudo-)random and a deterministic grid in combination with the support of

`e1`

.

##### References

Huber, P.J. (1981) *Robust Statistics*. New York: Wiley.

Rieder, H. (1994) *Robust Asymptotic Statistics*. New York: Springer.

##### See Also

`ContaminationSize`

, `TotalVarDist`

,
`HellingerDist`

, `Distribution-class`

##### Examples

```
# NOT RUN {
KolmogorovDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
mixCoeff=c(0.2,0.8)))
KolmogorovDist(Norm(), Td(10))
KolmogorovDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100))
KolmogorovDist(Pois(10), Binom(size = 20))
KolmogorovDist(Norm(), rnorm(100))
KolmogorovDist((rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5), Norm())
KolmogorovDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))
# }
```

*Documentation reproduced from package distrEx, version 2.8.0, License: LGPL-3*