# CvMDist

##### Generic function for the computation of the Cramer - von Mises distance of two distributions

Generic function for the computation of the Cramer - von Mises distance \(d_\mu\) 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 Cramer - von Mises distance is defined as $$d_\mu(P,Q)^2=\int\,(P(\{y\in\R^m\,|\,y\le x\})-Q(\{y\in\R^m\,|\,y\le x\}))^2\,\mu(dx)$$ where \(\le\) is coordinatewise on \(\R^m\).

- Keywords
- distribution

##### Usage

```
CvMDist(e1, e2, ...)
# S4 method for UnivariateDistribution,UnivariateDistribution
CvMDist(e1, e2, mu = e1, useApply = FALSE, ..., diagnostic = FALSE)
# S4 method for numeric,UnivariateDistribution
CvMDist(e1, e2, mu = e1, ..., diagnostic = FALSE)
```

##### Arguments

- e1
object of class

`"Distribution"`

or class`"numeric"`

- e2
object of class

`"Distribution"`

- …
further arguments to be used e.g. by

`E()`

- useApply
logical; to be passed to

`E()`

- mu
object of class

`"Distribution"`

; integration measure; defaulting to`e2`

- diagnostic
logical; if

`TRUE`

, the return value obtains an attribute`"diagnostic"`

with diagnostic information on the integration, i.e., a list with entries`method`

(`"integrate"`

or`"GLIntegrate"`

),`call`

,`result`

(the complete return value of the method),`args`

(the args with which the method was called), and`time`

(the time to compute the integral).

##### Details

Diagnostics on the involved integrations are available if argument
`diagnostic`

is `TRUE`

. Then there is attribute `diagnostic`

attached to the return value, which may be inspected
and accessed through `showDiagnostic`

and
`getDiagnostic`

.

##### Value

Cramer - von Mises distance of `e1`

and `e2`

##### Methods

- e1 = "UnivariateDistribution", e2 = "UnivariateDistribution":
Cramer - von Mises distance of two univariate distributions.

- e1 = "numeric", e2 = "UnivariateDistribution":
Cramer - von Mises distance between the empirical formed from a data set (e1) and a univariate distribution.

##### References

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

##### See Also

`ContaminationSize`

, `TotalVarDist`

,
`HellingerDist`

, `KolmogorovDist`

,
`Distribution-class`

##### Examples

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

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