TotalVarDist {distrEx} | R Documentation |
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,{\cal A})
.
The total variation distance is defined as
d_v(P,Q)=\sup_{B\in{\cal A}}|P(B)-Q(B)|
TotalVarDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
TotalVarDist(e1,e2,
rel.tol=.Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'numeric,DiscreteDistribution'
TotalVarDist(e1, e2, ...)
## S4 method for signature 'DiscreteDistribution,numeric'
TotalVarDist(e1, e2, ...)
## S4 method for signature 'numeric,AbscontDistribution'
TotalVarDist(e1, e2, asis.smooth.discretize = "discretize",
n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
rel.tol = .Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ...,
diagnostic = FALSE)
## 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$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ...,
diagnostic = FALSE)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
TotalVarDist(e1, e2,
rel.tol = .Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
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 – (in package distrEx: just
used for distributions with a.c. parts, where it is used to pass on arguments
to |
diagnostic |
logical; if |
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.
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
.
Total variation distance of e1
and e2
total variation distance of two absolutely continuous
univariate distributions which is computed using distrExIntegrate
.
total variation distance of absolutely continuous and discrete
univariate distributions (are mutually singular; i.e.,
have distance =1
).
total variation distance of two discrete univariate distributions
which is computed using support
and sum
.
total variation distance of discrete and absolutely continuous
univariate distributions (are mutually singular; i.e.,
have distance =1
).
Total variation distance between (empirical) data and a discrete distribution.
Total variation distance between (empirical) data and a discrete distribution.
Total variation distance between (empirical) data and an abs. cont. distribution.
Total variation distance between (empirical) data and an abs. cont. distribution.
Total variation distance of mixed discrete and absolutely continuous univariate distributions.
Matthias Kohl Matthias.Kohl@stamats.de,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de
Huber, P.J. (1981) Robust Statistics. New York: Wiley.
Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.
TotalVarDist-methods
, ContaminationSize
,
KolmogorovDist
, HellingerDist
,
Distribution-class
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))