Generic function for the computation of the Hellinger distance \(d_h\) of two distributions \(P\) and \(Q\) which may be defined for an arbitrary sample space \((\Omega,{\cal A})\). The Hellinger distance is defined as $$d_h(P,Q)=\frac{1}{2}\int|\sqrt{dP}\,-\sqrt{dQ}\,|^2$$ where \(\sqrt{dP}\), respectively \(\sqrt{dQ}\) denotes the square root of the densities.
HellingerDist(e1, e2, ...)
# S4 method for AbscontDistribution,AbscontDistribution
HellingerDist(e1,e2,
rel.tol=.Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ...)
# S4 method for AbscontDistribution,DiscreteDistribution
HellingerDist(e1,e2, ...)
# S4 method for DiscreteDistribution,AbscontDistribution
HellingerDist(e1,e2, ...)
# S4 method for DiscreteDistribution,DiscreteDistribution
HellingerDist(e1,e2, ...)
# S4 method for numeric,DiscreteDistribution
HellingerDist(e1, e2, ...)
# S4 method for DiscreteDistribution,numeric
HellingerDist(e1, e2, ...)
# S4 method for numeric,AbscontDistribution
HellingerDist(e1, e2, asis.smooth.discretize = "discretize",
n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
rel.tol=.Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ...)
# S4 method for AbscontDistribution,numeric
HellingerDist(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$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ...)
# S4 method for AcDcLcDistribution,AcDcLcDistribution
HellingerDist(e1,e2,
rel.tol=.Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ...)object of class "Distribution" or class "numeric"
object of class "Distribution" or class "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 (not in package distrEx)
Hellinger distance of e1 and e2
Hellinger distance of two absolutely continuous
univariate distributions which is computed using distrExintegrate.
Hellinger distance of absolutely continuous and discrete
univariate distributions (are mutually singular; i.e.,
have distance =1).
Hellinger distance of two discrete univariate distributions
which is computed using support and sum.
Hellinger distance of discrete and absolutely continuous
univariate distributions (are mutually singular; i.e.,
have distance =1).
Hellinger distance between (empirical) data and a discrete distribution.
Hellinger distance between (empirical) data and a discrete distribution.
Hellinger distance between (empirical) data and an abs. cont. distribution.
Hellinger distance between (empirical) data and an abs. cont. distribution.
Hellinger distance of mixed discrete and absolutely continuous univariate distributions.
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 Hellinger 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.
Huber, P.J. (1981) Robust Statistics. New York: Wiley.
Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.
distrExIntegrate, ContaminationSize,
TotalVarDist, KolmogorovDist,
Distribution-class
# NOT RUN {
HellingerDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
mixCoeff=c(0.2,0.8)))
HellingerDist(Norm(), Td(10))
HellingerDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100)) # mutually singular
HellingerDist(Pois(10), Binom(size = 20))
x <- rnorm(100)
HellingerDist(Norm(), x)
HellingerDist(x, Norm(), asis.smooth.discretize = "smooth")
y <- (rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5)
HellingerDist(y, Norm())
HellingerDist(y, Norm(), asis.smooth.discretize = "smooth")
HellingerDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))
# }
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