# HellingerDist

0th

Percentile

##### Generic function for the computation of the Hellinger distance of two distributions

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.

Keywords
distribution
##### Usage
HellingerDist(e1, e2, ...)
# S4 method for AbscontDistribution,AbscontDistribution
HellingerDist(e1,e2,
rel.tol=.Machine$double.eps^0.3, TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ..., diagnostic = FALSE) # S4 method for AbscontDistribution,DiscreteDistribution HellingerDist(e1,e2, ...) # S4 method for DiscreteDistribution,AbscontDistribution HellingerDist(e1,e2, ...) # S4 method for DiscreteDistribution,DiscreteDistribution HellingerDist(e1,e2, ...) # S4 method for numeric,DiscreteDistribution HellingerDist(e1, e2, ...) # S4 method for DiscreteDistribution,numeric HellingerDist(e1, e2, ...) # S4 method for numeric,AbscontDistribution HellingerDist(e1, e2, asis.smooth.discretize = "discretize", n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2), up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"), rel.tol=.Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
# 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$double.eps^0.3, TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ..., diagnostic = FALSE) # S4 method for AcDcLcDistribution,AcDcLcDistribution HellingerDist(e1,e2, rel.tol=.Machine$double.eps^0.3,
TruncQuantile = getdistrOption("TruncQuantile"),
IQR.fac = 15, ..., diagnostic = FALSE)
##### Arguments
e1

object of class "Distribution" or class "numeric"

e2

object of class "Distribution" or class "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.

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 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).

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.

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

Hellinger distance of e1 and e2

##### Methods

e1 = "AbscontDistribution", e2 = "AbscontDistribution":

Hellinger distance of two absolutely continuous univariate distributions which is computed using distrExintegrate.

e1 = "AbscontDistribution", e2 = "DiscreteDistribution":

Hellinger distance of absolutely continuous and discrete univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":

Hellinger distance of two discrete univariate distributions which is computed using support and sum.

e1 = "DiscreteDistribution", e2 = "AbscontDistribution":

Hellinger distance of discrete and absolutely continuous univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "numeric", e2 = "DiscreteDistribution":

Hellinger distance between (empirical) data and a discrete distribution.

e1 = "DiscreteDistribution", e2 = "numeric":

Hellinger distance between (empirical) data and a discrete distribution.

e1 = "numeric", e2 = "AbscontDistribution":

Hellinger distance between (empirical) data and an abs. cont. distribution.

e1 = "AbscontDistribution", e1 = "numeric":

Hellinger distance between (empirical) data and an abs. cont. distribution.

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":

Hellinger distance of mixed discrete and absolutely continuous univariate distributions.

##### References

Huber, P.J. (1981) Robust Statistics. New York: Wiley.

Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.

distrExIntegrate, ContaminationSize, TotalVarDist, KolmogorovDist, Distribution-class

##### Aliases
• HellingerDist
• HellingerDist-methods
• HellingerDist,AbscontDistribution,AbscontDistribution-method
• HellingerDist,AbscontDistribution,DiscreteDistribution-method
• HellingerDist,DiscreteDistribution,DiscreteDistribution-method
• HellingerDist,DiscreteMVDistribution,DiscreteMVDistribution-method
• HellingerDist,DiscreteDistribution,AbscontDistribution-method
• HellingerDist,LatticeDistribution,DiscreteDistribution-method
• HellingerDist,DiscreteDistribution,LatticeDistribution-method
• HellingerDist,LatticeDistribution,LatticeDistribution-method
• HellingerDist,numeric,DiscreteDistribution-method
• HellingerDist,DiscreteDistribution,numeric-method
• HellingerDist,numeric,AbscontDistribution-method
• HellingerDist,AbscontDistribution,numeric-method
• HellingerDist,AcDcLcDistribution,AcDcLcDistribution-method
##### Examples
# 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))
# }

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

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