# AsymTotalVarDist

##### Generic function for the computation of asymmetric total variation distance of two distributions

Generic function for the computation of asymmetric total variation distance \(d_v(\rho)\) of two distributions \(P\) and \(Q\) where the distributions may be defined for an arbitrary sample space \((\Omega,{\cal A})\). For given ratio of inlier and outlier probability \(\rho\), this distance is defined as $$d_v(\rho)(P,Q)=\int (dQ-c\,dP)_+$$ for \(c\) defined by $$\rho \int (dQ-c\,dP)_+ = \int (dQ-c\,dP)_-$$ It coincides with total variation distance for \(\rho=1\).

- Keywords
- distribution

##### Usage

```
AsymTotalVarDist(e1, e2, ...)
# S4 method for AbscontDistribution,AbscontDistribution
AsymTotalVarDist(e1,e2, rho = 1,
rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
# S4 method for AbscontDistribution,DiscreteDistribution
AsymTotalVarDist(e1,e2, rho = 1, ...)
# S4 method for DiscreteDistribution,AbscontDistribution
AsymTotalVarDist(e1,e2, rho = 1, ...)
# S4 method for DiscreteDistribution,DiscreteDistribution
AsymTotalVarDist(e1,e2, rho = 1, ...)
# S4 method for numeric,DiscreteDistribution
AsymTotalVarDist(e1, e2, rho = 1, ...)
# S4 method for DiscreteDistribution,numeric
AsymTotalVarDist(e1, e2, rho = 1, ...)
# S4 method for numeric,AbscontDistribution
AsymTotalVarDist(e1, e2, rho = 1, asis.smooth.discretize = "discretize",
n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
# S4 method for AbscontDistribution,numeric
AsymTotalVarDist(e1, e2, rho = 1,
asis.smooth.discretize = "discretize",
n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e1),
up.discr = getUp(e1), h.smooth = getdistrExOption("hSmooth"),
rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
# S4 method for AcDcLcDistribution,AcDcLcDistribution
AsymTotalVarDist(e1, e2,
rho = 1, rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
```

##### Arguments

- e1
object of class

`"Distribution"`

or`"numeric"`

- e2
object of class

`"Distribution"`

or`"numeric"`

- asis.smooth.discretize
possible methods are

`"asis"`

,`"smooth"`

and`"discretize"`

. Default is`"discretize"`

.- n.discr
if

`asis.smooth.discretize`

is equal to`"discretize"`

one has to specify the number of lattice points used to discretize the abs. cont. distribution.- low.discr
if

`asis.smooth.discretize`

is equal to`"discretize"`

one has to specify the lower end point of the lattice used to discretize the abs. cont. distribution.- up.discr
if

`asis.smooth.discretize`

is equal to`"discretize"`

one has to specify the upper end point of the lattice used to discretize the abs. cont. distribution.- h.smooth
if

`asis.smooth.discretize`

is equal to`"smooth"`

-- i.e., the empirical distribution of the provided data should be smoothed -- one has to specify this parameter.- rho
ratio of inlier/outlier radius

- rel.tol
relative tolerance for

`distrExIntegrate`

and`uniroot`

- maxiter
parameter for

`uniroot`

- Ngrid
How many grid points are to be evaluated to determine the range of the likelihood ratio?

- TruncQuantile
Quantile the quantile based integration bounds (see details)

- IQR.fac
Factor for the scale based integration bounds (see details)

- …
further arguments to be used in particular methods -- (in package distrEx: just used for distributions with a.c. parts, where it is used to pass on arguments to

`distrExIntegrate`

).- 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

For distances between absolutely continuous distributions, we use numerical
integration; to determine sensible bounds we proceed as follows:
by means of `min(getLow(e1,eps=TruncQuantile),getLow(e2,eps=TruncQuantile))`

,
`max(getUp(e1,eps=TruncQuantile),getUp(e2,eps=TruncQuantile))`

we determine
quantile based bounds `c(low.0,up.0)`

, and by means of
`s1 <- max(IQR(e1),IQR(e2));`

`m1<- median(e1);`

`m2 <- median(e2)`

and `low.1 <- min(m1,m2)-s1*IQR.fac`

, `up.1 <- max(m1,m2)+s1*IQR.fac`

we determine scale based bounds; these are combined by
`low <- max(low.0,low.1)`

, `up <- max(up.0,up1)`

.

Again in the absolutely continuous case, to determine the range of the
likelihood ratio, we evaluate this ratio on a grid constructed as follows:
```
x.range <- c(seq(low, up, length=Ngrid/3),
q.l(e1)(seq(0,1,length=Ngrid/3)*.999),
q.l(e2)(seq(0,1,length=Ngrid/3)*.999))
```

Finally, for both discrete and absolutely continuous case,
we clip this ratio downwards by `1e-10`

and upwards by `1e10`

In case we want to compute the total variation distance between (empirical) data
and an abs. cont. distribution, we can specify the parameter `asis.smooth.discretize`

to avoid trivial distances (distance = 1).

Using `asis.smooth.discretize = "discretize"`

, which is the default,
leads to a discretization of the provided abs. cont. distribution and
the distance is computed between the provided data and the discretized
distribution.

Using `asis.smooth.discretize = "smooth"`

causes smoothing of the
empirical distribution of the provided data. This is, the empirical
data is convoluted with the normal distribution `Norm(mean = 0, sd = h.smooth)`

which leads to an abs. cont. distribution. Afterwards the distance
between the smoothed empirical distribution and the provided abs. cont.
distribution is computed.

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

Asymmetric Total variation distance of `e1`

and `e2`

##### Methods

- e1 = "AbscontDistribution", e2 = "AbscontDistribution":
total variation distance of two absolutely continuous univariate distributions which is computed using

`distrExIntegrate`

.- e1 = "AbscontDistribution", e2 = "DiscreteDistribution":
total variation distance of absolutely continuous and discrete univariate distributions (are mutually singular; i.e., have distance

`=1`

).- e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":
total variation distance of two discrete univariate distributions which is computed using

`support`

and`sum`

.- e1 = "DiscreteDistribution", e2 = "AbscontDistribution":
total variation distance of discrete and absolutely continuous univariate distributions (are mutually singular; i.e., have distance

`=1`

).- e1 = "numeric", e2 = "DiscreteDistribution":
Total variation distance between (empirical) data and a discrete distribution.

- e1 = "DiscreteDistribution", e2 = "numeric":
Total variation distance between (empirical) data and a discrete distribution.

- e1 = "numeric", e2 = "AbscontDistribution":
Total variation distance between (empirical) data and an abs. cont. distribution.

- e1 = "AbscontDistribution", e1 = "numeric":
Total variation distance between (empirical) data and an abs. cont. distribution.

- e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":
Total variation distance of mixed discrete and absolutely continuous univariate distributions.

##### References

to be filled; Agostinelli, C and Ruckdeschel, P. (2009): A simultaneous inlier and outlier model by asymmetric total variation distance.

##### See Also

`TotalVarDist-methods`

, `ContaminationSize`

,
`KolmogorovDist`

, `HellingerDist`

,
`Distribution-class`

##### Examples

```
# NOT RUN {
AsymTotalVarDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
mixCoeff=c(0.2,0.8)), rho=0.3)
AsymTotalVarDist(Norm(), Td(10), rho=0.3)
AsymTotalVarDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100), rho=0.3) # mutually singular
AsymTotalVarDist(Pois(10), Binom(size = 20), rho=0.3)
x <- rnorm(100)
AsymTotalVarDist(Norm(), x, rho=0.3)
AsymTotalVarDist(x, Norm(), asis.smooth.discretize = "smooth", rho=0.3)
y <- (rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5)
AsymTotalVarDist(y, Norm(), rho=0.3)
AsymTotalVarDist(y, Norm(), asis.smooth.discretize = "smooth", rho=0.3)
AsymTotalVarDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5), rho=0.3)
# }
```

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