Generic function for the computation of the total variation distance of two distributions
Description
Generic function for the computation of the total variation distance d_v of two distributions P and Q where the distributions may be defined for an arbitrary sample space (Omega, A). The total variation distance is defined as
d_v(P,Q)=\sup{P(B)Q(B)  B in A}
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33  TotalVarDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
TotalVarDist(e1,e2,
rel.tol=.Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ...)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'numeric,DiscreteDistribution'
TotalVarDist(e1, e2, ...)
## S4 method for signature 'DiscreteDistribution,numeric'
TotalVarDist(e1, e2, ...)
## S4 method for signature 'numeric,AbscontDistribution'
TotalVarDist(e1, e2, asis.smooth.discretize = "discretize",
n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
rel.tol = .Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ...)
## S4 method for signature 'AbscontDistribution,numeric'
TotalVarDist(e1, e2, asis.smooth.discretize = "discretize",
n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e1),
up.discr = getUp(e1), h.smooth = getdistrExOption("hSmooth"),
rel.tol = .Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ...)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
TotalVarDist(e1, e2,
rel.tol = .Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ...)

Arguments
e1 
object of class 
e2 
object of class 
asis.smooth.discretize 
possible methods are 
n.discr 
if 
low.discr 
if 
up.discr 
if 
h.smooth 
if 
rel.tol 
relative accuracy requested in integration 
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 (not in package distrEx) 
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)
.
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.
Value
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
andsum
.  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.
Author(s)
Matthias Kohl Matthias.Kohl@stamats.de,
Peter Ruckdeschel peter.ruckdeschel@unioldenburg.de
References
Huber, P.J. (1981) Robust Statistics. New York: Wiley.
Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.
See Also
TotalVarDistmethods
, ContaminationSize
,
KolmogorovDist
, HellingerDist
,
Distributionclass
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  TotalVarDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
mixCoeff=c(0.2,0.8)))
TotalVarDist(Norm(), Td(10))
TotalVarDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100)) # mutually singular
TotalVarDist(Pois(10), Binom(size = 20))
x < rnorm(100)
TotalVarDist(Norm(), x)
TotalVarDist(x, Norm(), asis.smooth.discretize = "smooth")
y < (rbinom(50, size = 20, prob = 0.5)10)/sqrt(5)
TotalVarDist(y, Norm())
TotalVarDist(y, Norm(), asis.smooth.discretize = "smooth")
TotalVarDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))
