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 AbscontDistribution,AbscontDistribution
TotalVarDist(e1,e2,
rel.tol=.Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
# S4 method for AbscontDistribution,DiscreteDistribution
TotalVarDist(e1,e2, ...)
# S4 method for DiscreteDistribution,AbscontDistribution
TotalVarDist(e1,e2, ...)
# S4 method for DiscreteDistribution,DiscreteDistribution
TotalVarDist(e1,e2, ...)
# S4 method for numeric,DiscreteDistribution
TotalVarDist(e1, e2, ...)
# S4 method for DiscreteDistribution,numeric
TotalVarDist(e1, e2, ...)
# S4 method for numeric,AbscontDistribution
TotalVarDist(e1, e2, asis.smooth.discretize = "discretize",
n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
rel.tol = .Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ...,
diagnostic = FALSE)
# S4 method for 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$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ...,
diagnostic = FALSE)
# S4 method for AcDcLcDistribution,AcDcLcDistribution
TotalVarDist(e1, e2,
rel.tol = .Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)Total variation distance of e1 and e2
object of class "Distribution" or "numeric"
object of class "Distribution" or "numeric"
possible methods are "asis",
"smooth" and "discretize". Default is "discretize".
if asis.smooth.discretize is equal to
"discretize" one has to specify the number of lattice points
used to discretize the abs. cont. distribution.
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.
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.
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.
relative accuracy requested in integration
Quantile the quantile based integration bounds (see details)
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).
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).
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
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.
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))
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