# Chiral Anomaly in interacting Condensed Matter Systems

Colin Rylands  Joint Quantum Institute and Condensed Matter Theory Center,
University of Maryland, College Park, MD 20742, USA  Alireza Parhizkar  Joint
Quantum Institute and Condensed Matter Theory Center, University of Maryland,
College Park, MD 20742, USA  Anton A. Burkov  Department of Physics and
Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
Victor Galitski  Joint Quantum Institute and Condensed Matter Theory Center,
University of Maryland, College Park, MD 20742, USA

###### Abstract

The chiral anomaly is a fundamental quantum mechanical phenomenon which is of
great importance to both particle physics and condensed matter physics alike.
In the context of QED it manifests as the breaking of chiral symmetry in the
presence of electromagnetic fields. It is also known that anomalous chiral
symmetry breaking can occur through interactions alone, as is the case for
interacting one dimensional systems. In this paper we investigate the
interplay between these two modes of anomalous chiral symmetry breaking in the
context of interacting Weyl semimetals. Using Fujikawa’s path integral method
we show that the chiral charge continuity equation is modified by the presence
of interactions which can be viewed as including the effect of the electric
and magnetic fields generated by the interacting quantum matter. This can be
understood further using dimensional reduction and a Luttinger liquid
description of the lowest Landau level. These effects manifest themselves in
the non-linear response of the system. In particular we find an interaction
dependent density response due to a change in the magnetic field as well as a
contribution to the non-equilibrium and inhomogeneous anomalous Hall response
while preserving its equilibrium value.

Introduction—Modern condensed matter physics has benefited greatly from
concepts originally introduced in the context of high energy physics. One such
concept is the chiral anomaly; the breaking of classical chiral symmetry in a
quantum theory Adler (1969); Bell and Jackiw (1969). Within QED it arises
through the need to regularize certain loop diagrams which contain differences
of linearly divergent integrals. The appropriate regularization can either
preserve charge conservation symmetry, chiral symmetry or some combination of
the two but not both. On physical grounds, the first of these is chosen, which
brings about a source term for the divergence of the chiral current,
$j_{5}^{\mu}$, whenever electric and magnetic fields are not orthogonal,

$\partial_{\mu}j^{\mu}_{5}=\frac{e^{2}}{2\pi^{2}}\mathbf{E}\cdot\mathbf{B}.$
(1)

Here $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic fields and we
have set $c=\hbar=1$. This expression, although derived from a single triangle
diagram in perturbation theory was shown to obey non-renormalization theorems;
higher order terms cannot modify the form of this equation and are accounted
for by replacing the bare fields and charge with their renormalized values
Adler and Bardeen (1969). Later, this was reinforced when it was discovered
that the chiral anomaly manifests in the path integral formalism through the
non-invariance of the measure under a chiral symmetry transformation Fujikawa
(1979, 1980); Fujikawa and Suzuki (2004).

The chiral anomaly is present for all odd spatial dimensions Frampton and
Kephart (1983a, b); Zumino  _et al._ (1984) but is particularly important in
one spatial dimension where it is crucial for the proper treatment of
interacting fermionic theories through bosonization Naón (1985); Lee and Chen
(1988). A prominent feature therein is that chiral symmetry breaking can occur
due to the presence of interactions even when electromagnetic fields are
absent. Indeed, it is well known, although perhaps not expressed in this way,
that the chiral charge conservation equation for interacting fermions is
Giamarchi (2003); Gogolin  _et al._ (2004)

$\displaystyle\partial_{\mu}j_{5}^{\mu}=\frac{\lambda^{2}}{2\pi}\partial_{1}j_{5}^{1},$
(2)

where $\lambda^{2}/2$ is the strength of the density-density interactions and
the index $1$ refers to the spatial direction. By writing the expression in
this form we have separated out the part which appears due to the non-
invariance of the path integral measure. If an electric field is present also,
it will appear as an additional $eE/\pi$ term on the right hand side Nielsen
and Ninomiya (1983).

Chiral symmetry, is an emergent low energy property in condensed matter
systems, appearing due to an even number of chiral modes crossing the Fermi
surface which are actually part of the same band. In this respect, the anomaly
can be understood in non-interacting systems via the pumping of chiral charge
through the bottom of the band from one node to another Nielsen and Ninomiya
(1983). Despite not being a fundamental symmetry, it is intimately related to
many key concepts including the quantized Hall conductance, e.g. through
Laughlins’s argument Laughlin (1981), and more recently the existence of
topological metals such as the Weyl semimetal Wan  _et al._ (2011); Burkov and
Balents (2011); Yang  _et al._ (2011); Xu  _et al._ (2011); Halász and Balents
(2012); Aji (2012); Weng  _et al._ (2015); Lv  _et al._ (2015a, b); Xu  _et
al._ (2015); Huang  _et al._ (2015). In this Letter we examine the interplay
between the two modes of chiral symmetry breaking expressed through (1) and
(2) in the context of interacting condensed matter systems. Specifically, we
show that for short range interactions the anomaly can be written as

$\displaystyle\partial_{\mu}j^{\mu}_{5}=\frac{e^{2}}{2\pi^{2}}\tilde{\mathbf{E}}\cdot\tilde{\mathbf{B}},$
(3)

where $\tilde{\mathbf{E}}$ and $\tilde{\mathbf{B}}$, defined below, contain
the effect of both the electromagnetic fields in a manner similar to (1) and
the interactions through terms like in (2).

The effect of interactions in Weyl semimetals has been considered previously
using perturbative means Jho and Kim (2013); Gorbar  _et al._ (2013, 2014);
Miransky and Shovkovy (2015); Maciejko and Nandkishore (2014). In contrast,
our work takes a non-perturbative approach and considers the interactions from
the outset through the chiral anomaly itself. By utilizing (3) we predict a
number of new non-perturbative phenomena found beyond linear response which
can be expected in interacting Weyl semimetals and attributed to the chiral
anomaly.

Model—We consider a model of interacting Dirac fermions, $\psi$, in the
presence of a constant background magnetic field in $3+1$ dimensions. The
action is $S=S_{0}+S_{\text{int}}$ with

$\displaystyle S_{0}=\int
d^{4}x\,\bar{\psi}(x)\left[i\not{\partial}+e\not{A}\right]\psi(x),$ (4)

where we have employed Dirac slash notation and
$\bar{\psi}=\psi^{\dagger}\gamma_{0}$. For later convenience we split the
gauge field, $A^{\mu}=A^{\mu}_{0}+\tilde{A}^{\mu}$, into a part describing the
magnetic field pointing along the $\hat{z}$ direction,
$A^{\mu}_{0}=xB_{z}\delta^{\mu}_{2}$ and a perturbation around it,
$\tilde{A}^{\mu}$. The magnetic field breaks the Lorentz invariance down to
rotational invariance in the transverse plane spanned by the $\hat{x}$ and
$\hat{y}$ directions and reduced $(1+1)$-d Lorentz symmetry in the
longitudinal directions. The general short range current-current interaction
is of the form

$S_{\text{int}}=-\frac{1}{2}\int
d^{4}x\,\lambda^{2}_{\mu\nu}j^{\mu}(x)j^{\nu}(x),$ (5)

where $j^{\mu}(x)=\bar{\psi}(x)\gamma^{\mu}\psi(x)$ is the fermion current
with $\lambda^{2}_{\mu\nu}=\lambda_{\mu\alpha}\lambda_{\nu}^{\ \alpha}$ being
the interaction strength. Although certain results presented in this Letter
are more general, we have restricted our focus to special cases of
$\lambda^{2}_{\mu\nu}=\lambda^{2}\eta_{\mu\nu}$, which preserves Lorentz
symmetry,
$\lambda^{2}_{\mu\nu}=\lambda^{2}_{0}\eta_{0\mu}\eta_{0\nu}+\lambda^{2}_{3}\eta_{3\mu}\eta_{3\nu}$
which preserves the reduced symmetries of our system if
$\lambda^{2}_{0}=\lambda^{2}_{3}$ and which gives density-density interaction
when $\lambda^{2}_{3}=0$ 111$\eta_{\mu\nu}$ is the metric of the space-time
which for our purposes here is considered to be flat. Also,
$\delta^{\mu}_{\nu}$ is the Kronecker delta. Throughout the Letter we use
Einstein’s notations and sometimes represent the four-vector of current as
$j^{\mu}\equiv(\rho,j^{x},j^{y},j^{z})$ in Minkowski coordinates
$x^{\mu}\equiv(t,x,y,z)$. . Evidently, depending on the choice of
$\lambda_{\mu\nu}$ some of the symmetries of the model may be broken, e.g.
Lorentz invariance, but they do not break the classical chiral symmetry. These
interactions are RG irrelevant and typically are not considered, however we
will see that in the presence of the constant magnetic field, they should not
be discounted.

Chiral Anomaly & Interactions—To study the chiral anomaly in the presence of
interactions we proceed using a generalization of Fujikawa’s path integral
method Fujikawa (1979, 1980). The path integral is

$I=\int\mathcal{D}\left[\bar{\psi}\psi a_{\mu}\right]\exp{i\Big{\\{}\int
d^{4}x\,\bar{\psi}i\not{D}\psi+\frac{1}{2}a_{\mu}a^{\mu}}\Big{\\}},$ (6)

where we have introduced the Hubbard-Stratonovich field $a_{\mu}(x)$ which has
been included in the generalized Dirac operator as
$D_{\mu}=\partial_{\mu}-ieA_{\mu}-i\lambda_{\mu\nu}a^{\nu}$, and whose
equation of motion reads $a_{\mu}=-\lambda_{\nu\mu}j^{\nu}$. Integration over
the auxiliary $a_{\mu}$ field gives the original action
$S=S_{0}+S_{\text{int}}$ back. We now perform an infinitesimal chiral
transformation, $\psi\to
e^{i\theta(x)\gamma_{5}}\psi,~{}\bar{\psi}\to\bar{\psi}e^{i\theta(x)\gamma_{5}}$
which results in a shift of the action,

$\displaystyle S\to S+\int
d^{4}x\,\theta(x)\left[\partial_{\mu}j^{\mu}_{5}-\mathcal{A}_{5}(x)\right],$
(7)

where $j^{\mu}_{5}(x)=\bar{\psi}(x)\gamma^{\mu}\gamma_{5}\psi(x)$ is the
chiral current. The first term in the brackets arises from the classical shift
of the action itself whereas the second is the anomalous term which is a
result of the non-invariance of the measure. It takes the standard form
$\mathcal{A}_{5}(x)=2\text{Tr}{\left[\theta(x)\gamma_{5}\right]}$ or more
explicitly

$\displaystyle\mathcal{A}_{5}(x)=2\theta(x)\sum_{n}\varphi^{\dagger}_{n}(x)\gamma_{5}\,\varphi_{n}(x),$
(8)

where $\varphi_{n}(x)$ are some orthonormal basis of wavefunctions used to
expand the Grassmann variables $\psi(x)=\sum c_{n}\phi_{n}(x)$. In the absence
of interactions the natural choice is to take these to be the eigenfunctions
of $\not{D}_{0}=\not{\partial}-ie\not{A}$ and regularize this divergent sum
using the heat kernel method $\sum_{n}\to\lim_{M\to
0}\sum_{n}e^{-\not{D}^{2}_{0}/M}$. Such a choice of basis has the crucial
benefit of formally diagonalizing the action. This results in the familiar
anomalous term
$\mathcal{A}_{5}(x)=\theta(x)\frac{e^{2}}{16\pi^{2}}F_{\mu\nu}F_{\rho\sigma}\epsilon^{\mu\nu\rho\sigma}$
with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$\epsilon^{\mu\nu\rho\sigma}$ is the Levi-Cevita symbol. The chiral anomaly
equation (1) then follows. Note that owing to the fact that
$\\{\gamma_{5},\not{D}_{0}\\}=0$ it is evident from (8) that anomalous term is
generated solely by the zero modes of the Dirac operator.

In the presence of interactions we regularize the sum using the generalized
Dirac operator, including the Hubbard-Stratonovich field
$\not{D}=\gamma^{\mu}(\partial_{\mu}-ieA_{\mu}-i\lambda_{\mu\nu}a^{\nu})~{}$(For
similar approaches see [33; 34]). Following the same procedure we find,
$\mathcal{A}_{5}(x)=\theta(x)\frac{1}{16\pi^{2}}\mathcal{F}_{\mu\nu}\mathcal{F}_{\rho\sigma}\epsilon^{\mu\nu\rho\sigma}$
where
$\mathcal{F}_{\mu\nu}=\partial_{\mu}(eA_{\nu}+\lambda_{\nu\alpha}a^{\alpha})-\partial_{\nu}(eA_{\mu}+\lambda_{\mu\beta}a^{\beta})$
and after integrating over $a_{\mu}$ we find

$\displaystyle\partial_{\mu}j^{\mu}_{5}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{16\pi^{2}}F_{\mu\nu}F_{\rho\sigma}\epsilon^{\mu\nu\rho\sigma}-\frac{e}{2\pi^{2}}\epsilon^{\mu\nu\rho\sigma}\lambda^{2}_{\sigma\alpha}\partial_{\mu}A_{\nu}\partial_{\rho}j^{\alpha}$
(9)
$\displaystyle+\frac{1}{4\pi^{2}}\epsilon^{\mu\nu\rho\sigma}\lambda^{2}_{\nu\alpha}\lambda^{2}_{\sigma\beta}\partial_{\mu}j^{\alpha}\partial_{\rho}j^{\beta}.$

We see that there are terms depending only on the electromagnetic field, only
on the presence of interactions and a mixed term requiring the presence of
both. After defining

$\displaystyle\tilde{E}_{i}$ $\displaystyle=$ $\displaystyle
E_{i}-\frac{1}{e}\left[\lambda^{2}_{i\beta}\partial_{0}-\lambda^{2}_{0\beta}\partial_{i}\right]j^{\beta},$
(10) $\displaystyle\tilde{B}_{i}$ $\displaystyle=$ $\displaystyle
B_{i}-\frac{1}{2e}\epsilon_{ijk}\left[\lambda^{2}_{j\beta}\partial_{k}-\lambda^{2}_{k\beta}\partial_{j}\right]j^{\beta},$
(11)

equation (3) is obtained.

We could view this as a screening, by the interactions of the electric and
magnetic fields which are responsible for the non-conservation of the chiral
charge. This can be seen more clearly by allowing the electromagnetic fields
to be dynamical and, for simplicity, considering
$\lambda^{2}_{\mu\nu}=\lambda^{2}\eta_{\mu 0}\eta_{\nu 0}$, i.e. density-
density interactions. Upon treating the electromagnetic field in a semi-
classical fashion through $ej^{\nu}=\partial_{\mu}F^{\nu\mu}$, we find that
$\tilde{\mathbf{E}}=\mathbf{E}-\frac{\lambda^{2}}{e^{2}}\mathbf{\nabla}\left(\mathbf{\nabla}\cdot\mathbf{E}\right)$
and $\tilde{\mathbf{B}}=\mathbf{B}$. Therefore the anomalous chiral symmetry
breaking is generated not only by the background fields but also by the
fluctuations induced by the interacting matter.

Dimensional reduction to a Luttinger liquid— The chiral anomaly, in the free
case, can be straightforwardly understood through dimensional reduction of the
$(3+1)$-d system to the $(1+1)$-d linearly dispersing lowest Landau level
(LLL) via a magnetic field, $B_{z}$ Nielsen and Ninomiya (1983). We show now
that one can also arrive at (3) using dimensional reduction provided that the
LLL is described by a Luttinger liquid.

Let us consider a system that is homogeneous in the transverse directions
along $\hat{x}$ and $\hat{y}$. In particular, the only external fields are in
the longitudinal $\hat{z}$ direction and there are no currents which vary
along $\hat{x}$ and $\hat{y}$. Our anomalous relation then reduces to

$\partial_{\mu}j^{\mu}_{5}=\frac{e^{2}}{8\pi^{2}}F_{\mu\nu}F_{\rho\sigma}\epsilon^{\mu\nu\rho\sigma}-\frac{eB_{z}}{2\pi^{2}}\lambda^{2}_{\sigma\alpha}\epsilon^{12\rho\sigma}\partial_{\rho}j^{\alpha}.$
(12)

Assuming that the interacting system still forms Landau levels, the zero modes
which are responsible for the anomaly are present only on the LLL. As in the
free case, the magnetic field achieves a dimensional reduction from the
$(3+1)$-d theory to the LLL which is effectively $(1+1)$-d. Within the LLL the
following identity is valid
$\epsilon^{12\rho\sigma}\gamma_{\sigma}=\gamma_{5}\gamma^{\rho}$ and after
some rearranging we arrive at

$\partial_{\mu}j^{\mu}_{5}=\frac{1}{1+n_{0}\lambda^{2}_{3}/\pi}\frac{e^{2}}{2\pi^{2}}E_{z}B_{z}-\frac{n_{0}\left(\lambda^{2}_{0}-\lambda^{2}_{3}\right)/\pi}{1+n_{0}\lambda^{2}_{3}/\pi}\partial_{3}j^{3}_{5},$
(13)

where $n_{0}=\frac{eB_{z}}{2\pi}$. Here we have also specialized to the case
where the interaction tensor is diagonal. In deriving this equation we have
assumed that Landau levels are formed in the interacting system or more
precisely that there is a spin polarized LLL on which the anomaly is
generated. We have made no assumptions on the nature of Landau levels or how
they arise, only that they exist which seems a physically reasonable
proposition especially in the limit of large background field. In the opposite
limit of zero background field (13) reduces to the noninteracting result.

The second term in (13) is similar to (2) while the modification of the first
has been discovered before in early studies of interacting $(1+1)$-d fermions
Georgi and Rawls (1971); Shei (1972). To understand their appearance better we
introduce the following action consisting of $N$ coupled $(1+1)$-d bosonic
fields

$\displaystyle S$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\int\frac{d^{2}x}{2\pi}\Bigg{\\{}\left[\partial_{t}\phi_{j}\right]^{2}+\left[\partial_{x}\phi_{j}\right]^{2}-e\left[\epsilon^{mn}A_{m}\partial_{n}\right]\phi_{j}$
(14) $\displaystyle+\sum_{j\leq
k}\frac{\lambda^{2}_{0}}{\pi}[\partial_{x}\phi_{j}][\partial_{x}\phi_{k}]+\frac{\lambda^{2}_{3}}{\pi}[\partial_{t}\phi_{j}][\partial_{t}\phi_{k}]\Bigg{\\}},$

with $\epsilon^{mn}$ the $2$-d Levi-Cevita symbol. This is equivalent, through
bosonization, to a system of $N$ flavors of interacting chiral fermions,
$\chi^{\dagger}_{\pm,j}=\sqrt{\rho_{0}}e^{i\left[\pm\phi_{j}-\int^{t}dt\partial_{x}\phi_{j}\right]}$
where $\rho_{0}$ is the background density Giamarchi (2003); Gogolin  _et al._
(2004). The bosons are related to the fermionic charge and chiral charge
density via
$\sum_{\sigma=\pm}:\chi^{\dagger}_{\sigma,j}\chi_{\sigma,j}:=-\partial_{x}\phi_{j}/\pi$
and
$\sum_{\sigma=\pm}\sigma:\chi_{\sigma,j}^{\dagger}\chi_{\sigma,j}:=\partial_{t}\phi_{j}/\pi$
with $:\,:$ indicating normal ordering.

The model is flavor symmetric and accordingly both the interactions and the
gauge field affect only the symmetric combination,
$\phi_{S}=\frac{1}{\sqrt{N}}\sum_{j}\phi_{j}$. After a canonical
transformation and retaining only the symmetric terms we arrive at the
following action

$\displaystyle S_{S}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}x}{2\pi}\left(1+\lambda_{0}^{2}N/\pi\right)[\partial_{x}\phi_{S}]^{2}+\left(1+\lambda_{3}^{2}N/\pi\right)[\partial_{t}\phi_{S}]^{2}$
(15)
$\displaystyle-2\sqrt{N}eA_{0}\partial_{x}\phi_{S}+2\sqrt{N}eA_{3}\partial_{t}\phi_{S}.$

Note that here the gauge field couples to the fermionic density rather than
through minimal coupling with the symmetric boson, an important distinction
which we comment on further below. The chiral anomaly is now manifest in the
Euler-Lagrange equation for $\phi_{S}$. Calculating this we find agreement
with (13) provided one identifies the number of flavors with the Landau level
degeneracy, $N=n_{0}=eB_{z}/2\pi$ as well as
$j_{5}^{0}=\sum\partial_{t}\phi_{j}/\pi$ and
$j_{5}^{3}=\sum\partial_{x}\phi_{j}/\pi$ which follows from the properties of
$\gamma^{\mu}$ in $(1+1)-$d.

Our path integral calculation is therefore consistent with a description of
the LLL as a Luttinger liquid. A Luttinger liquid approach has also been
adopted in [37] to investigate the effect of disorder which we shall not
consider here. The Luttinger liquid consists of a pair of interacting chiral
fermions
$\chi^{\dagger}_{\pm,S}=\sqrt{\rho_{0}}e^{i\left[\pm\phi_{S}-\int^{t}dt\partial_{x}\phi_{S}\right]}$
formed from the symmetric boson which couple to the gauge field and the
decoupled non-symmetric fields which play no role. The excitations of the LLL
are still chiral but are distinct from these bare fermions and are created by
$\Psi^{\dagger}_{\pm}=\sqrt{\rho_{0}}e^{i\left[\pm\sqrt{1+\lambda_{0}^{2}N/\pi}\phi_{S}-\sqrt{1+\lambda_{3}^{2}N/\pi}\int^{t}dt\partial_{x}\phi_{S}\right]}$
which coincide with $\chi^{\dagger}_{\pm,S}$ only when interactions are
absent. In general these excitations carry different electric and chiral
charges from $\chi^{\dagger}_{\pm,s}$ which can be seen through the
coefficients of $\phi_{S}$ and $\int^{t}dt\partial_{x}\phi_{S}$ in the
exponential. Had our gauge field coupled to these instead then we would find
that the chiral anomaly equation was unmodified. A similar situation also
arises when comparing conductances in one dimensional systems Alekseev  _et
al._ (1996).

As mentioned in the introduction, the chiral anomaly is related to Laughlin’s
argument for quantized Hall conductance Laughlin (1981). Therein one can argue
that the invariance of the Hall conductance to local interactions implies
invariance of the chiral anomaly for the edge modes of Laughlin’s cylinder and
vice versa. We remark that our results are not in contradiction to this as our
$(1+1)$-d chiral modes are not spatially separated as they are in Laughlin’s
argument. In order to see similar interaction effects as ours one would need
to include non-local interactions between the edges.

Consequences for Weyl Semimetals —We now turn our attention to the
consequences of (3) for interacting interacting condensed matter systems, in
particular Weyl semimetals. These are a recently discovered type of gapless
topological matter possessing a number of distinctive features which arise due
to the chiral anomaly including a large negative magnetoresistance Nielsen and
Ninomiya (1983); Son and Spivak (2013); Burkov (2014); Fukushima  _et al._
(2008) and an anomalous Hall response Zyuzin and Burkov (2012); Chen  _et al._
(2013). The low energy description of such systems is given by
$S=S_{0}+S_{b}+S_{\text{int}}$ with $S_{b}=\int d^{4}x\,b_{\mu}j_{5}^{\mu},$
where $b_{\mu}$ separates the Weyl nodes in momentum and energy space. The
effect of this term is most conveniently seen by performing a chiral rotation
$\psi\to
e^{ib_{\mu}x^{\mu}\gamma_{5}}\psi,~{}\bar{\psi}\to\bar{\psi}e^{ib_{\mu}x^{\mu}\gamma_{5}}$
which removes $S_{b}$ at the cost of generating a Chern-Simons term, $S_{CS}$
due to the chiral anomaly. In terms of the Hubbard-Stratonovich field this is

$\displaystyle
S_{CS}=\int\frac{d^{4}x}{4\pi^{2}}\epsilon^{\nu\mu\rho\sigma}b_{\mu}\left[eA_{\nu}+\lambda_{\nu\alpha}a^{\alpha}\right]\partial_{\rho}\left[eA_{\sigma}+\lambda_{\sigma\beta}a^{\beta}\right].$
(16)

Then, following [34] we vary $S+S_{CS}$ with respect to $A_{1}$ to obtain the
anomalous Hall current. Specializing to the case
$b_{\mu}=b_{z}\delta^{3}_{\mu}$, $\lambda_{\mu\nu}=\lambda\eta_{\mu\nu}$ and
after integrating over $a_{\mu}$ we find
$j^{x}=\frac{eb_{z}}{2\pi^{2}}\tilde{E}^{y}$ or more expicitly

$\displaystyle
j^{x}=\frac{eb_{z}}{2\pi^{2}}E^{y}-\frac{\lambda^{2}b_{z}}{2\pi^{2}}\left[\partial_{t}j^{y}-\partial_{y}\rho\right],$
(17)

with $E^{y}$ being the electric field along $\hat{y}$ and $\rho(x)=j^{0}(x)$.
The first term here gives the quantum anomalous Hall current while the
interaction dependent contribution vanishes in equilibrium. Thus, the
interactions do not affect the equilibrium Hall current however they may
contribute to the non-equilibrium or inhomogeneous response. Combining (17)
with the corresponding expression for $j^{y}$ and switching to Fourier space
we obtain the homogeneous finite frequency Hall conductivity expected from
$S_{CS}$,

$\displaystyle\sigma^{xy}(\omega)=\left[1+\left(\frac{\lambda^{2}b_{z}}{2\pi^{2}}\omega\right)^{2}\right]^{-1}\frac{e^{2}b_{z}}{2\pi^{2}}.$
(18)

The effect of interactions can also be seen in the equilibrium density
response to a change in the magnetic field, $B_{z}\to B_{z}+\delta B_{z}$. In
the absence of any fields along the transverse components we may use (13) as
our anomalous relation. After subtracting the background density, the leading
order density response is

$\displaystyle\delta
j^{0}=\frac{1}{1+\lambda^{2}\frac{eB_{z}}{2\pi^{2}}}\frac{eb_{z}}{2\pi^{2}}\delta
B_{z}.$ (19)

Due to the dimensional reduction, the density is equivalent to a chiral
current in the longitudinal direction,
$\left<j^{0}\right>=\left<j^{3}_{5}\right>$ and so (19) can be viewed as the
generation of a chiral current in response to a change in the magnetic field
which is known as the chiral separation effect (CSE) Vilenkin (1980);
Metlitski and Zhitnitsky (2005); Newman and Son (2006).

Photon Action—As was pointed out in Chen  _et al._ (2013) the Chern-Simons
term obtained via chiral transformation requires some subtle interpretation if
it is to describe a Weyl semimetal. The appropriate understanding comes from
integrating out the fermionic degrees of freedom to determine the linear
response. We adopt this approach to confirm the equilibrium response of the
system expected from $S_{CS}$. To $\mathcal{O}(e^{2})$, after integrating out
the fermions,

$\displaystyle S$ $\displaystyle=$
$\displaystyle-e\int\frac{d^{3}qd\omega}{(2\pi)^{4}}\text{Tr}\left[G_{\lambda}(\mathbf{q},\omega)\gamma^{\mu}\right]\tilde{A}^{*}_{\mu}(\mathbf{q},\omega)$
$\displaystyle-\frac{e^{2}}{2}\int\frac{d^{3}qd\omega}{(2\pi)^{4}}\tilde{A}_{\mu}(\mathbf{q},\omega)\Pi_{\lambda}^{\mu\nu}(\mathbf{q},\omega)\tilde{A}^{*}_{\nu}(\mathbf{q},\omega),$
(20)

where $G_{\lambda}(\mathbf{q},\omega)$ is the single particle, interacting,
Green’s function in the presence of $B_{z}$ and $b_{z}$ and
$\Pi_{\lambda}^{\mu\nu}(\mathbf{q},\omega)=\int\frac{d^{3}q^{\prime}d\omega^{\prime}}{(2\pi)^{4}}\text{Tr}\left[\gamma^{\mu}G_{\lambda}(\mathbf{q^{\prime}},\omega^{\prime})\gamma^{\nu}G_{\lambda}(\mathbf{q^{\prime}}-\mathbf{q},\omega^{\prime}-\omega)\right]$.
The anomalous terms we are interested in can then be isolated by considering
the leading $\mathbf{q},\omega\to 0$ terms which provide the static
homogeneous response.

The evaluation of $G_{\lambda}(\mathbf{q},\omega)$ cannot be carried out
exactly however we are only interested in computing the density response and
the form of (19) suggestive of an RPA approximation. Indeed, the low energy
response in the longitudinal directions is determined solely by the LLL whose
current and density responses are completely captured by an RPA summation
owing to its reduced dimensionality. Using the non-interacting Green’s
function in the Landau level basis derived in [30] we obtain

$\lim_{\begin{subarray}{c}\mathbf{q}\to 0\\\ \omega\to
0\end{subarray}}\Pi^{\mu\nu}_{\text{RPA}}(\mathbf{q},\omega)=\left[\frac{1}{1+\lambda^{2}\frac{eB_{z}}{2\pi^{2}}}P_{\parallel}+P_{\perp}\right]^{\mu}_{\rho}\lim_{\begin{subarray}{c}\mathbf{q}\to
0\\\ \omega\to 0\end{subarray}}\Pi^{\rho\nu}_{0}(\mathbf{q},\omega),$ (21)

where, for $\lambda^{2}_{\mu\nu}=\lambda^{2}\eta_{\mu\nu}$,
$P_{\parallel}=(1-\gamma_{3})/2$ projects onto the longitudinal components,
while $P_{\perp}=1-P_{\parallel}$ projects onto the transverse components.
When $\lambda^{2}_{\mu\nu}=\lambda^{2}\eta_{0\mu}\eta_{0\nu}$ we use instead
$P_{\parallel}=[(1-\gamma_{3})/2][(1-\gamma_{5})/2]$ which projects only onto
the temporal components. We see here a screening of the density response due
to the interactions while the transverse components are unaffected. The
equilibrium Hall response is therefore the same as the free case, in agreement
with (17). The linear density response is then found after computing
$\raisebox{2.15277pt}{\scalebox{0.8}{$\displaystyle\lim_{\mathbf{q}\to
0}\;$}}\raisebox{2.15277pt}{\scalebox{0.8}{$\displaystyle\lim_{\omega\to
0}\;$}}\Pi^{02}_{0}(\mathbf{q},\omega)/iq_{x}$. Surprisingly however, this
vanishes. Thus the anomalous density response comes from the first term in
(20) and can be attributed to the change in degeneracy of the LLL. The same
RPA screening occurs for this term also and we find agreement with (19).

In the absence of $B_{z}$, the density response depends on all filled bands
Chen  _et al._ (2013). When it is present however, this is not the case and
the density response is determined only by the LLL. Therefore we can
understand this by returning to our description of the LLL given in (15). The
$S_{b}$ term can be accounted for by the inclusion of a chemical potential
term $S_{S,b}=-\int d^{2}x\sqrt{N}b_{z}\partial_{x}\phi_{S}/\pi$. Recalling
that $N=eB_{z}/2\pi$ is identified with the degeneracy of the LLL we compute
the density response to $N\to N+\delta N$ and once again find agreement with
(19). Furthermore, the modification of the anomalous terms is natural from
this viewpoint as we can identify $(1+\lambda^{2}eB_{z}/2\pi^{2})^{-1}$ as
being the charge susceptibility or the chiral charge stiffness of the LLL
Giamarchi (2003); Gogolin  _et al._ (2004). This is in agreement with (19)
being viewed either as the density response or the CSE.

Conclusions— In this Letter we have explored the interplay between anomalous
chiral symmetry breaking via electromagnetic fields and interactions. We have
shown, using Fujikawa’s path integral method, that the chiral charge
continuity equation contains new interaction dependent terms which can be
absorbed into effective electromagnetic fields which are responsible for the
breaking of chiral symmetry. Furthermore this result was shown to be
consistent with the lowest Landau level being a Luttinger liquid. We
investigated the consequences of this result for interacting Weyl semimetals
and found that interaction effects will be present in the non-equilibrium Hall
response as well as the density response to a change in the magnetic field.
These results were then reproduced via direct perturbative calculation.

Recently, it was discovered that the circular photogalvanic effect de Juan
_et al._ (2017), originally thought to be quantized as a result of the chiral
anomaly, is actually renormalized due to the presence of interactions
Avdoshkin  _et al._ (2020). It would be desirable to understand our results in
the context of this observable also. Lastly, we note that other anomalous Ward
identities, including the gravitational anomaly can be derived using
Fujikawa’s method and our analysis can likewise be applied in those situations
with the possibility of additional observable interaction effects Gooth  _et
al._ (2017).

###### Acknowledgements.

We acknowledge useful discussions with Natan Andrei. This work was supported
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences
under Award No. DE-SC0001911, the Simons Foundation (A.P., C.R., and V.G.) and
Natural Sciences and Engineering Research Council (NSERC) of Canada (AAB).
Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science and Economic Development
and by the Province of Ontario through the Ministry of Economic Development,
Job Creation and Trade.

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Centrality Estimators for Probability Density Functions

Djemel Ziou
---
Département d’informatique
Université de Sherbrooke
Sherbrooke, Qc., Canada J1K 2R1
<EMAIL_ADDRESS>

###### Abstract

In this report, we explore the data selection leading to a family of
estimators maximizing a centrality. The family allows a nice properties
leading to accurate and robust probability density function fitting according
to some criteria we define. We establish a link between the centrality
estimator and the maximum likelihood, showing that the latter is a particular
case. Therefore, a new probability interpretation of Fisher maximum likelihood
is provided. We will introduce and study two specific centralities that we
have named Hölder and Lehmer estimators. A numerical simulation is provided
showing the effectiveness of the proposed families of estimators opening the
door to development of new concepts and algorithms in machine learning, data
mining, statistics, and data analysis.

Keywords: Data selection, Centrality estimator, Hölder estimator, Lehmer
estimator, maximum likelihood, histogram fitting.

##  1 Introduction

Maximum likelihood estimator (MLE) is a general statistical estimation method
proposed by R. A. Fisher almost one century before. Since then, many writings
have been published about the MLE, its variants, its properties, its
interpretation, and the controversy it provokes. In the case of iid data,
Fisher defined the probability of $X=\\{x_{1},\cdots,x_{n}\\}$ occurring as
being proportional to the probability
$P_{\theta}(X)=\prod_{i}^{n}P_{\theta}(x_{i})$. For the discrete random
variable $x$, the probability $P_{\theta}(x_{i})=P_{\theta}(x=x_{i})$. In the
continuous case, assuming that the iid data $X$ are sampled from a probability
density function (PDF) $h(x|\theta)$ with a parameter $\theta$, a value
$x_{i}$ is assumed to be known within some accuracy of measurement $\epsilon$.
The probability that $y$ falls in the cell
$[x_{i}-\epsilon/2,x_{i}+\epsilon/2]$ is
$P_{\theta}(x_{i})=\int_{y\in[x_{i}-\epsilon/2,x_{i}+\epsilon/2]}h(y|\theta)dy$.
Let assume that the cell width $\epsilon$ is small and the PDF $h(y|\theta)$
is almost constant within the interval $[x_{i}-\epsilon/2,x_{i}+\epsilon/2]$,
it follows that $P_{\theta}(x_{i})\simeq\epsilon h(x_{i}|\theta)$. Whatever
$x$ is discrete or continuous random variable, the most probable set of values
for $\theta$ will make the probability $P_{\theta}(X)$ maximal.

Let us take things from a different perspective. Our motivation is caused on
the one hand, by errors in the real data, the iid hypothesis, the unknown PDF,
the size of the cell $\epsilon$. The errors in the data can have different
origins including the measuring instruments, the human, the unavailability of
the measurements. It is often difficult to formally sustain that the data are
assumed to be generated from a known PDF or they are iid. In practice, the iid
assumption is at the basis of the likelihood definition and the chosen PDF is
retained if it leads to acceptable clustering, segmentation, prediction or
decision. In the case of a continuous random variable, the cell size
$\epsilon$ is considered as a constant, with no effect on the maximum
likelihood estimate. The transition from the probability to a PDF is valid if
the cell size is chosen adequately. The information theory and statistical
model selection made it clear that the cell size depends on the observed
Fisher information [14]. In addition to validity of transition from
probability to PDF, the cell size $\epsilon$ is of great importance because it
informs about the amount of required data for the PDF fitting.

On the other side, the likelihood as the product of probabilities
$\prod_{i}^{n}P_{\theta}(x_{i})$ is somehow ”harsh” as it may be highly
corrupted by a single low outlier. Considering that the root $n^{th}$ of this
product is a geometric mean indicates that $\prod_{i}^{n}P_{\theta}(x_{i})$ is
closer to the smallest values of $P_{\theta}(x_{i})$. However, accepting that
the most probable values for the $\theta$’s is observed on any cell over the
support covered by $X$ leads to consider $P_{\theta}(X)$ as a centrality
measure of $P_{\theta}(x_{i})$. Among the many existing centralities, we
choose those of Hölder and Lehmer, given respectively by:

$H_{\theta}(X;\alpha)=(\sum_{i=1}^{n}\lambda_{i}P_{\theta}^{\alpha}(x_{i}))^{1/\alpha}$
(1)
$L_{\theta}(X,\alpha)=\frac{\sum_{i=1}^{n}\lambda_{i}P_{\theta}^{\alpha}(x_{i})}{\sum_{i=1}^{n}\lambda_{i}P_{\theta}^{\alpha-1}(x_{i})}$
(2)

where the non-negative weights fulfills $\sum_{i=1}^{n}\lambda_{i}=1$. Each of
Hölder and Lehmer centralities implement specific data selection criteria [5].
Note that the uses in data selection, estimation and modeling in various
fields of these centralities and Pythagorean means which are special cases are
given in [4, 5]. The centralities has nice properties allowing to overcome the
above mentioned issue by choosing the $\alpha$ leading to a higher probability
$P_{\theta}(X)$. The estimation can be more accurate, according to some
criteria, at the cost of increased complexity. From now on, as in the
likelihood case, we are not interested on the probability of the data
$P_{\theta}(X)$, rather our focus is the parameter $\theta$. This
generalization of the likelihood have several advantages: 1) it overcomes the
IID assumption; 2) thanks to embedded data selection criterion in each of the
centralities, the maximum of a centrality (which we call centrality estimator
and we denote C-estimator), if exists, can be less sensitive to errors in
data. However, this generalization leads to new issues and a better
understanding of Fisher’s proposal. The properties of $P_{\alpha}(X)$, the
existence of the maximum, Fisher information, and the selection of the
suitable centrality are among issues.

Our goal is not to provide a rigorous mathematical reasoning, but rather to
deal with the computation of C-estimators, their properties and their
evaluation, under the following assumptions: 1) the random variable is
considered continuous and the values of the PDF
$h(x_{1}|\theta),\cdots,h(x_{n}|\theta)$ are not all equal as it is the case
of the uniform distribution; 2) The centralities in Eqs. 1 and 2 are positive,
continuous and derivable w.r.t to both $\alpha$ and $\theta$. The next two
sections are dedicated respectively to the study of Hölder and Lehmer
centralities. The sections 2 and 3 describe Hölder and Lehmer centrality
measures and some of their properties. A comparison of the two measures is
described in section 4. Section 5 is devoted to the maximum centrality
estimation. The accuracy of the C-estimators is explained in section 6. A case
study is the subject of the section 7.

##  2 Hölder measure

In this work, we consider the continuous random variable case where the cell
size is function of the Fisher information as it is explained before;
$P_{\theta}(y_{i})\simeq\epsilon(\theta)h(y_{i}|\theta)$. In this case, the
centrality, denoted H-Centrality (H-C) when using Eq. 1 is rewritten as
follow:

$H_{\alpha}(\theta)=(\sum_{i}\lambda_{i}(\epsilon(\theta)h(x_{i}|\theta))^{\alpha})^{1/\alpha}$
(3)

where $\alpha\in\mathbb{R}$. We assume that $H_{\alpha}(\theta)>0$ is
continuous and derivable w.r.t to both $\alpha$ and $\theta\in\Theta$ and
$\epsilon(\theta)>0$ is continuous and derivable w.r.t to $\theta$. An
interpretation of this equation is as follows: a data element
$h(x_{i}|\theta)$ is transformed to $h^{\alpha}(x_{i}|\theta)$, then the
arithmetic mean is computed in the transformed space and it is back
transformed to the original space. As it will be explained further, the
transform and its inverse implement data selection criteria. We will now state
properties of $H_{\alpha}(\theta)$.

P1: $lim_{\alpha\to
0}H_{\alpha}(\theta)=\prod_{i}(\epsilon(\theta)h(x_{i}|\theta))^{\lambda_{i}}$.
The limit of $\lim_{\alpha\to 0}lnH_{\alpha}(\theta)=0/0$, so we use the
Hôpital’s rule to get $\lim_{\alpha\to 0}lnH_{\alpha}(\theta)=\lim_{\alpha\to
0}(\sum_{i}ln((\epsilon(\theta)h(x_{i}|\theta))^{\lambda_{i}})exp(\alpha
ln(\epsilon(\theta)h(x_{i}|\theta)))/\sum_{i}\lambda_{i}(\epsilon(\theta)h(x_{i}|\theta))^{\alpha})=\sum_{i}ln((\epsilon(\theta)h(x_{i}|\theta)))^{\lambda_{i}}$.
Note that, the H-C $lim_{\alpha\to 0}H_{\alpha}(\theta)$ is the geometric mean
and it is the Fisher’s likelihood to the power $1/n$. It follows that the
Fisher’s likelihood inherits from the geometric mean the following feature: it
is dominated by smallest values of the set
{$\epsilon(\theta)h(x_{1}|\theta),\cdots,\epsilon(\theta)h(x_{n}|\theta)$}.

P2: $H_{\alpha}(\theta)$ reduces to the arithmetic mean when $\alpha=1$, and
Harmonic mean when $\alpha=-1$. This is straightforward from Eq. 3.

P3: The function $H_{\alpha}(\theta)$ is increasing in $\alpha$. In the case
of the Hölder mean, there are several proofs based on Hölder, Jensen, and
Schwartz inequalities, theory of convex functions, and critical point theory
[27, 16, 24, 25]. They are too long and require advanced mathematical
concepts. We provide here a simplest proof using elementary calculus in the
case of $H_{\alpha}(\theta)$ in Eq. 3. Let us consider the sign of the
derivative of $H_{\alpha}(\theta)$ in $\alpha$:

$\frac{\partial
H_{\alpha}(\theta)}{\partial\alpha}=\frac{H_{\alpha}(\theta)}{\alpha}(\sum_{i}\lambda_{i}(\frac{h(x_{i}|\theta)}{H_{\alpha}(\theta)})^{\alpha}lnh(x_{i}|\theta)-lnH_{\alpha})$
(4)

Because
$\sum_{i}\lambda_{i}(\frac{h(x_{i}|\theta)}{H_{\alpha}(\theta)})^{\alpha}=1$,
so

$\frac{\partial
H_{\alpha}(\theta)}{\partial\alpha}=\frac{H_{\alpha}(\theta)}{\alpha}\sum_{i}\lambda_{i}(\frac{h(x_{i}|\theta)}{H_{\alpha}(\theta)})^{\alpha}(lnh(x_{i}|\theta)-lnH_{\alpha}(\theta))$
(5)

Equivalently,

$\frac{\partial
H_{\alpha}(\theta)}{\partial\alpha}=\frac{H_{\alpha}(\theta)}{\alpha}\sum_{i}\lambda_{i}(\frac{h(x_{i}|\theta)}{H_{\alpha}(\theta)})^{\alpha}ln\frac{h(x_{i}|\theta)}{H_{\alpha}(\theta)}$
(6)

Because the term $z_{i}=(h(x_{i}|\theta)/H_{\alpha}(\theta))^{\alpha}>0$, then
$z_{i}>ln(z_{i})$ and therefore the lower band of the derivative of
$H_{\alpha}(\theta)$ is straightforward:

$\frac{\partial
H_{\alpha}(\theta)}{\partial\alpha}>H_{\alpha}(\theta)\sum_{i}\lambda_{i}(ln\frac{h(x_{i}|\theta)}{H_{\alpha}(\theta)})^{2}>0$
(7)

It follows that $H_{\alpha}(\theta)$ is an increasing function; that is there
is no critical point $\alpha$ of $\frac{\partial
H_{\alpha}(\theta)}{\partial\alpha}$. Consequently, the choice of $\alpha$ is
an open issue that deserves further study.

P4: If the PDF $h(x|\theta)$ is bounded then $H_{\alpha}(\theta)$ is also a
bounded function. Indeed, $H_{\alpha}(\theta)$ is mean of
$\epsilon(\theta)h(x_{1}|\theta)\leq\cdots\leq\epsilon(\theta)h(x_{n}|\theta)$,
so $\epsilon(\theta)h(x_{1}|\theta)\leq
H_{\alpha}(\theta)\leq\epsilon(\theta)h(x_{n}|\theta)$ by definition of the
mean. Moreover,
$lim_{\alpha\rightarrow+\infty}H_{\alpha}(\theta)=lim_{\alpha\rightarrow+\infty}\epsilon(\theta)h(x_{n}|\theta)(\sum_{i}\lambda_{i}(h(x_{i}|\theta)/h(x_{n}|\theta))^{\alpha})^{1/\alpha}=\epsilon(\theta)h(x_{n}|\theta)$.
Similarly,
$lim_{\alpha\rightarrow-\infty}H_{\alpha}(\theta)=lim_{\alpha\rightarrow+\infty}\epsilon(\theta)h(x_{1}|\theta)(\sum_{i}\lambda_{i}(h(x_{i}|\theta)/h(x_{1}|\theta))^{-\alpha})^{1/-\alpha}=\epsilon(\theta)h(x_{1}|\theta)$.

P5: The H-C $H_{\alpha}(\theta)$ is a bijective function in $\alpha$. Indeed,
the inverse function theorem asserts that strictly increasing continuous
functions are bijective and therefore invertible in the interval
$[min_{i}h(x_{i}|\theta),max_{i}h(x_{i}|\theta)]$.

P6: The data $X$ and therefore the PDF values
$h(x_{1}|\theta),\cdots,h(x_{n}|\theta)$ do not have the same relevance in
$H_{\alpha}(\theta)$. Indeed, there are three data selection strategies
embedded in $H_{\alpha}(\theta)$. The first is the weight $\lambda_{i}>0$
encoding the relative relevance of the $i^{th}$ data element; i.e
$\sum_{i}\lambda_{i}=1$. The second is materialized by transform $\alpha$; for
a data element $x$, its relevance $h^{\alpha}(x|\theta)$ is increasing or
decreasing depending $h(x|\theta)$ itself. The $1/\alpha$ transform does not
cancel the $\alpha$ transform effects. The centrality $H_{\alpha}(\theta)$ is
increasing or decreasing depending on $H^{\alpha}_{\alpha}(\theta)$. The third
strategy concerns the influence of a PDF value $h(x_{i}|\theta)$ itself on the
$H_{\alpha}(\theta)$. The reader can find more about the data selection
strategies embedded in Pythagorean, Hölder, and Lehmer means in [4, 5].

P7: For a fixed $\theta$, the H-C $H_{\alpha}(\theta)$ is a probability of an
observation falling within a cell $s\mp\epsilon(\theta)/2$. Indeed, according
to P3,
$H_{\alpha}(\theta)\in[min_{i}\epsilon(\theta)h(x_{i}|\theta),max_{i}\epsilon(\theta)h(x_{i}|\theta)]$.
Because the probability $\epsilon(\theta)h(x|\theta)$ is assumed to be
continuous in $x$, it follows that there is at least one cell
$s\mp\epsilon(\theta)/2$ of $x$ such that
$H_{\alpha}(\theta)\simeq\epsilon(\theta)h(s|\theta)\in[min_{i}\epsilon(\theta)h(x_{i}|\theta),max_{i}\epsilon(\theta)h(x_{i}|\theta)]$.

P8: For a fixed $\theta$, the cell $s\mp\epsilon(\theta)/2$ is unique if
$h(x|\theta)$ is a bijective function in $x$. Indeed, if $h(x|\theta)$ is a
bijective function in $x$, then the function inversion theorem ensures that
there exists a unique $s=h^{-1}(H_{\alpha}(\theta)/\epsilon(\theta))$. If
$h(x|\theta)$ is only surjective, there are $m\in]1,n[$ cells
${s_{1},s_{2},\cdots,s_{m}}$ such that
$h(s_{1}|\theta)=...=h(s_{m}|\theta)=H_{\alpha}(\theta)/\epsilon(\theta)$. It
is worth noting that for a fixed $\theta$ and given data $X$, the parameter
$\alpha$ defines completely the H-C and the cell $s$. To make the dependencies
explicit, in the following, the center of the cell will be denoted
$s_{\alpha,\theta}$.

P9: For a fixed $\theta$, let $\alpha_{1}\leq\alpha_{2}$, we have
$h(s_{\alpha_{1},\theta}|\theta)\leq h(s_{\alpha_{2},\theta}|\theta)$. This is
straightforward because according to P3, we have
$H_{\alpha_{1}}(\theta)=\epsilon(\theta)h(s_{\alpha_{1},\theta})<H_{\alpha_{1}}(\theta)=\epsilon(\theta)h(s_{\alpha_{2},\theta})$
and $\epsilon(\theta)>0$. It follows that
$h(s_{\alpha_{1},\theta})<h(s_{\alpha_{2},\theta})$.

P10: $\int_{-\infty}^{+\infty}H_{\alpha}(\theta)d\alpha=+\infty$. Indeed,
since $H_{\alpha}(\theta)\geq c=min_{i}\epsilon(\theta)h(x_{i}|\theta)>0$,
then $\int_{-\infty}^{+\infty}H_{\alpha}(\theta)d\alpha\geq
c\int_{-\infty}^{+\infty}d\alpha=+\infty$.

P11: The expectation of $H_{\alpha}(\theta)$ with regards to a PDF $p(\alpha)$
is given by
$\int_{-\infty}^{+\infty}p(\alpha)H_{\alpha}(\theta)d\alpha=\epsilon(\theta)\\\
\int_{-\infty}^{+\infty}p(\alpha)h(s_{\alpha,\theta}|\theta)d\alpha$. The
equality is inferred from P8, because a cell
$s_{\alpha,\theta}\mp\epsilon(\theta)/2$ is chosen among $m\geq 1$ cells; i.e.
$H_{\alpha}(\theta)=h(s_{\alpha,\theta}|\theta)$.

##  3 Lehmer measure

From Eq. 2, we define the Lehmer centrality (L-C) as follow:

$L_{\alpha}(\theta)=\epsilon(\theta)\frac{\sum_{i}\lambda_{i}h^{\alpha}(x_{i}|\theta)}{\sum_{i}\lambda_{i}h^{\alpha-1}(x_{i}|\theta)}$
(8)

As for H-C, the L-C is a mean of the probabilities of an observation falling
within a cell of size $\epsilon(\theta)$. The data selection mechanisms
embedded in L-C are similar to those embedded in H-C (see P6).

Q1: $L_{0}(\theta)$ and $L_{1}(\theta)$ are related the harmonic and
arithmetic mean of $h^{\alpha}(x_{i}|\theta),~{}i=1,n$ respectively. This is
straightforward from Eq. 8.

Q2: The L-C is an increasing function in $\alpha$. Indeed,

$\frac{\partial
L_{\alpha}(\theta)}{\partial\alpha}=\epsilon(\theta)\frac{\sum_{i}\sum_{j}\lambda_{i}\lambda_{j}h^{\alpha-1}(x_{i}|\theta)h^{\alpha-1}(x_{j}|\theta)r_{ij}}{2(\sum_{i}h^{\alpha-1}(x_{i}|\theta))^{2}}$
(9)

where
$r_{ij}=(h(x_{i}|\theta)-h(x_{j}|\theta))(lnh(x_{i}|\theta)-lnh(x_{j}|\theta))$.
Because the sign of $h(x_{i}|\theta)-h(x_{j}|\theta)$ and ln
$h(x_{i}|\theta)-lnh(x_{j}|\theta)$ is the same, then $r_{ij}\geq 0$. All the
other terms are positive and not all $r_{ij}$ are zeros. It follows that
$\frac{\partial L_{\alpha}(\theta)}{\partial\alpha}>0$.

Q3: P4 until P11 remains valid for the L-C.

##  4 Comparison of H-C and L-C

R1: H-C and L-C satisfy

$L_{\alpha}(\theta)\left\\{\begin{array}[]{l c r}>H_{\alpha}(\theta)&\mbox{if
$\alpha>1$}\\\ <H_{\alpha}(\theta)&\mbox{if $\alpha<1$}\\\
=H_{\alpha}(\theta)&\mbox{if $\alpha=1$}\end{array}\right.$ (10)

Indeed, the L-C can be rewritten as

$L_{\alpha}(\theta)=(\frac{H_{\alpha}(\theta)}{H_{\alpha-1}(\theta)})^{\alpha-1}H_{\alpha}(\theta)$
(11)

Because $H_{\alpha}(\theta)$ is increasing in $\alpha$, we conclude that
$(\frac{H_{\alpha}(\theta)}{H_{\alpha-1}(\theta)})^{\alpha-1}$ is greater
(resp. smaller) than one if $\alpha\in]1,+\infty[$ (resp.
$\alpha\in]-\infty,1[$). Substituting $\alpha=1$ in Eqs. 3 and 8 leads to
$H_{1}(\theta)=L_{1}(\theta)$. From P4 and Q3,
$H_{\alpha}(\theta)=L_{\alpha}(\theta)=max_{i}\epsilon(\theta)h(x_{i}|\theta)$
(resp.
$H_{\alpha}(\theta)=L_{\alpha}(\theta)=min_{i}\epsilon(\theta)h(x_{i}|\theta))$
for $\alpha\to\infty$ (resp. $\alpha\to-\infty$).

##  5 Maximum centrality estimation

Having defined the Hölder and Lehmer centralities $H_{\alpha}(\theta)$ and
$L_{\alpha}(\theta)$, we will now deal with the estimation of $\theta$, by
maximizing each of they. None of these measures has a maximum in $\alpha$
because they are increasing with respect to this variable. Consequently, for
now we assume that $\alpha$ is known. Moreover, let us recall that
$H_{\alpha}(\theta)$ and $L_{\alpha}(\theta)$ are assumed both derivable in
$\theta$. In what follows, for the sake of simplicity, the cell size
$\epsilon(\theta)$ is ignored and the logarithm of both measures are designed
by $C_{\alpha}(\theta)$ where $C=\ln(H)$ in the case of Hölder centrality and
$C=\ln(L)$ for Lehmer one.

C1: $C_{\alpha}(\theta)$ has maximum in $\theta$ if
$h(s_{\alpha,\theta}|\theta)$ has a maximum in $\theta$. Indeed,
$C_{\alpha}(\theta)=ln(h(s_{\alpha,\theta}|\theta))$ and the PDF is supposed
to have a maximum with respect to $\theta$. Note that $s_{\alpha,\theta}$
depends on the value of $C$ being either $H$ or $L$.

C2: Let us now, look for $\theta$ as a critical point of the C-centrality.

$C_{\alpha}(\theta)=\left\\{\begin{array}[]{l r }\ln
H_{\alpha}^{\alpha}(\theta)-lnH_{\alpha-1}^{\alpha-1}(\theta)&\mbox{if
$C=L$}\\\ \frac{1}{\alpha}\ln H_{\alpha}^{\alpha}(\theta)&\mbox{if $C=H$
}\end{array}\right.$ (12)

Note that, $H_{\alpha}^{\alpha}(\theta)|_{\alpha=0}=1$. The first order
condition w.r.t $\theta$ is:

$\frac{\partial{C_{\alpha}}(\theta)}{\partial\theta}=\left\\{\begin{array}[]{l
r }\frac{1}{H_{\alpha}^{\alpha}}\frac{\partial
H_{\alpha}^{\alpha}(\theta)}{\partial\theta}-\frac{1}{H_{\alpha-1}^{\alpha-1}(\theta)}\frac{\partial
H_{\alpha-1}^{\alpha-1}(\theta)}{\partial\theta}=0&\mbox{if $C=L$}\\\
\frac{1}{\alpha H_{\alpha}^{\alpha}(\theta)}\frac{\partial
H_{\alpha}^{\alpha}(\theta)}{\partial\theta}=0&\mbox{if
$C=H$}\end{array}\right.$ (13)

Substituting H-L in Eq. 3, we obtain:

$\frac{\partial{C_{\alpha}}(\theta)}{\partial\theta}=\left\\{\begin{array}[]{l
r }\alpha E_{g_{\alpha}}(\frac{\partial
lnh(x|\theta)}{\partial\theta})-(\alpha-1)E_{g_{\alpha-1}}(\frac{\partial
lnh(x|\theta)}{\partial\theta})=0&\mbox{if $C=L$}\\\
E_{g_{\alpha}}(\frac{\partial lnh(x|\theta)}{\partial\theta})=0&\mbox{if
$C=H$}\end{array}\right.$ (14)

where $E_{g_{\alpha}}(\frac{\partial
lnh(x|\theta)}{\partial\theta})=\sum_{i}\frac{\partial
lnh(x_{i}|\theta)}{\partial\theta}g_{\alpha}(x_{i}|\theta)$ and the PDF
$g_{\alpha}(x_{i}|\theta)=\lambda_{i}h^{\alpha}(x_{i}|\theta)/H_{\alpha}^{\alpha}(\theta)$.

C4: The critical points $\theta^{*}_{L}$ and $\theta^{*}_{H}$ of H-C and L-C
fulfills $\theta^{*}_{L}=\theta^{*}_{H}$ if and only if $\alpha=1$ or when
$\theta^{*}_{H}$ is a critical point of both $H_{\alpha}^{\alpha}(\theta)$ and
$H_{\alpha-1}^{\alpha-1}(\theta)$. This is straightforward from Eq. 14.

C5: The second order condition w.r.t $\theta$ is given by:

$\frac{\partial^{2}C_{\alpha}(\theta)}{\partial\theta^{2}}=\left\\{\begin{array}[]{l
r
}\frac{1}{H_{\alpha}^{\alpha}(\theta)}\frac{\partial^{2}H_{\alpha}^{\alpha}(\theta)}{\partial^{2}\theta}-(\frac{1}{H_{\alpha}^{\alpha}(\theta)}\frac{\partial
H_{\alpha}^{\alpha}(\theta)}{\partial\theta})^{2}-\frac{1}{H_{\alpha-1}^{\alpha-1}(\theta)}\frac{\partial^{2}H_{\alpha-1}^{\alpha-1}(\theta)}{\partial^{2}\theta}+(\frac{1}{H_{\alpha-1}^{\alpha-1}(\theta)}\frac{\partial
H_{\alpha-1}^{\alpha-1}(\theta)}{\partial\theta})^{2}&\mbox{if $C=L$}\\\
\frac{1}{\alpha}\frac{1}{H_{\alpha}^{\alpha}(\theta)}\frac{\partial^{2}H_{\alpha}^{\alpha}(\theta)}{\partial^{2}\theta}-\alpha(\frac{1}{\alpha
H_{\alpha}^{\alpha}(\theta)}\frac{\partial
H_{\alpha}^{\alpha}(\theta)}{\partial\theta})^{2}&\mbox{if
$C=H$}\end{array}\right.$ (15)

At the critical point $\theta$ (Eq. 13), the second derivative is rewritten:

$\frac{\partial^{2}C_{\alpha}(\theta)}{\partial\theta^{2}}=\left\\{\begin{array}[]{l
r
}\frac{1}{H_{\alpha}^{\alpha}(\theta)}\frac{\partial^{2}H_{\alpha}^{\alpha}(\theta)}{\partial^{2}\theta}-\frac{1}{H_{\alpha-1}^{\alpha-1}(\theta)}\frac{\partial^{2}H_{\alpha-1}^{\alpha-1}(\theta)}{\partial^{2}\theta}&\mbox{if
$C=L$}\\\ \frac{1}{\alpha
H_{\alpha}^{\alpha}(\theta)}\frac{\partial^{2}H_{\alpha}^{\alpha}(\theta)}{\partial^{2}\theta}&\mbox{if
$C=H$}\end{array}\right.$ (16)

We have

$\frac{\partial^{2}lnH_{\alpha}^{\alpha}(\theta)}{\partial\theta^{2}}=\frac{\frac{\partial^{2}H_{\alpha}^{\alpha}(\theta)}{\partial\theta^{2}}}{H_{\alpha}^{\alpha}(\theta)}-(\frac{\partial
lnH_{\alpha}^{\alpha}(\theta)}{\partial\theta})^{2}$ (17)

Substituting this equation in Eq. 16 and by using $\frac{\partial
lnH_{\alpha}^{\alpha}(\theta)}{\partial\theta}=\frac{\partial
lnH_{\alpha-1}^{\alpha-1}(\theta)}{\partial\theta}$ when C=L and
$\frac{\partial lnH_{\alpha}^{\alpha}(\theta)}{\partial\theta}=0$ when C=H,
the second derivative becomes:

$\frac{\partial^{2}C_{\alpha}(\theta)}{\partial\theta^{2}}=\left\\{\begin{array}[]{l
r
}\frac{\partial^{2}lnH_{\alpha}^{\alpha}(\theta)}{\partial\theta^{2}}-\frac{\partial^{2}lnH_{\alpha-1}^{\alpha-1}(\theta)}{\partial\theta^{2}}&\mbox{if
$C=L$}\\\
\frac{1}{\alpha}\frac{\partial^{2}lnH_{\alpha}^{\alpha}(\theta)}{\partial\theta^{2}}&\mbox{if
$C=H$}\end{array}\right.$ (18)

Straightforward manipulations leads to

$\frac{\partial^{2}C_{\alpha}(\theta)}{\partial\theta^{2}}=\left\\{\begin{array}[]{l
r }\alpha E_{g_{\alpha}}((\frac{\partial
lnh(x|\theta)}{\partial\theta})^{2})+E_{g_{\alpha}}(\frac{\partial^{2}lnh(x|\theta)}{\partial\theta^{2}})-(\alpha-1)E_{g_{\alpha-1}}((\frac{\partial
lnh(x|\theta)}{\partial\theta})^{2})-E_{g_{\alpha-1}}(\frac{\partial^{2}lnh(x|\theta)}{\partial\theta^{2}})&\mbox{if
$C=L$}\\\ \alpha E_{g_{\alpha}}((\frac{\partial
lnh(x|\theta)}{\partial\theta})^{2})+E_{g_{\alpha}}(\frac{\partial^{2}lnh(x|\theta)}{\partial\theta^{2}})&\mbox{if
$C=H$}\end{array}\right.$ (19)

In the case of H-C, the critical point is a maximum if
$E_{g_{\alpha}}(\frac{\partial^{2}lnh(x|\theta)}{\partial\theta^{2}})<-\alpha
E_{g_{\alpha}}((\frac{\partial lnh(x|\theta)}{\partial\theta})^{2})<0$ when
$\alpha\geq 0$ and
$E_{g_{\alpha}}(\frac{\partial^{2}lnh(x|\theta)}{\partial\theta^{2}})<-\alpha
E_{g_{\alpha}}((\frac{\partial lnh(x|\theta)}{\partial\theta})^{2})$ when
$\alpha<0$. In the case of L-C, the critical point is a maximum if
$\alpha(E_{g_{\alpha}}((\frac{\partial
lnh(x|\theta)}{\partial\theta})^{2})-E_{g_{\alpha-1}}(\frac{\partial
lnh(x|\theta)}{\partial\theta})^{2}))<E_{g_{\alpha-1}}(\frac{\partial^{2}lnh(x|\theta)}{\partial\theta^{2}})-E_{g_{\alpha}}(\frac{\partial^{2}lnh(x|\theta)}{\partial\theta^{2}})-E_{g_{\alpha-1}}((\frac{\partial
lnh(x|\theta)}{\partial\theta})^{2})$.

##  6 Accuracy of C-estimators

The maximum likelihood estimators are often studied according to Fisher’s
requirements which are consistency, efficiency and sufficiency. But instead of
the theoretical study, we propose to define easy-to-implement performance
measures for the evaluation of the accuracy of the distribution fit. Even
several are possible to define, in this study, we will limit ourselves to two
measures. For now, we assume that the parameter $\alpha$ is known.

###  6.1 Residuals

For a given observation $x_{i}$ and a model $\theta$, the residual can be
measured by a norm of the absolute value of the difference between the
observation frequency $d_{i}$ and its predicted value $h(x_{i}|\theta)$. For
all observations, the arithmetic mean of the residuals is often used [13] for
the evaluation of the C-estimator $\theta$. Rather, we propose the use of
Hölder and Lehmer mean families instead. Let’s start with Hölder with the
parameter $\beta\in\mathbb{R}$:

$e_{H}=(\sum_{i=1}^{n}\lambda_{i}|d_{i}-h(x_{i}|\theta)|^{\beta})^{1/\beta}$
(20)

When $\beta=2$, $e_{H}$ reduces to the famous square root of the mean square
errors. The residual error $e_{H}$ inherits the properties of the Hölder
centrality measure: it is increasing in $\beta$ and implements a data
selection mechanisms [5, 4]. Let us now approach the interpretation of $e_{H}$
in terms of probability. Assuming that the error $\eta=\lambda(d-h(x|\theta))$
can be sampled from a normal distribution $N(0,\sigma)$, then the pdf of
$|\eta|$ is a half-normal with mean $\sigma\sqrt{2/\pi}$ and a variance
$\sigma^{2}(1-2/\pi)$. The pdf of $|\eta|^{\beta}$ is straightforward; let
$Y=|\eta|^{\beta}$, $Pr(Y\leq y)=Pr(|\eta|^{\beta}\leq y)=Pr(|\eta|\leq
y^{1/\beta})=F_{|\eta|}(y^{1/\beta})$, where $F$ is the cumulative
distribution function of the half-normal. Now, let us consider the iid random
variables $Y_{1},\cdots,Y_{n}$, then according to the central limit theorem,
the pdf of $Z=\sum_{i}Y_{i}$ is normal distribution with the mean equals to
$nE(Y)$ and variance $nVar(Y)$. Because the variable $Z$ is non negative, we
then use the truncated normal distribution. Under this choice, the pdf of
$e_{H}$ is straightforward; let $\psi=Z^{1/\beta}$, $Pr(\psi\leq
z)=Pr(Z^{1/\beta}\leq z)=Pr(Z\leq z^{\beta})=F_{Z}(z^{\beta})$, where:

$f_{H}(z)=F_{Z}^{{}^{\prime}}(z^{\beta})=\beta
z^{\beta-1}g(z^{\beta}|nE(Y),nVar(Y))/(1/2+erf(E(Y)/Var(Y)))$ (21)

and $g$ is the normal PDF and $z\geq 0$.

For Lehmer with the parameter $\beta\in\mathbb{R}$, we define the residual
error as follow:

$e_{L}=\frac{\sum_{i}\lambda_{i}|d_{i}-h(x_{i}|\theta)|^{\beta}}{\sum_{i}\lambda_{i}|d_{i}-h(x_{i}|\theta)|^{\beta-1}}$
(22)

The residual error $e_{L}$ inherits the properties of the Lehmer centrality
measure: it is increasing in $\beta$ and implements data selection mechanisms
[5, 4]. The random variable $Z$ associated with the error measure $e_{L}$ is
the ratio of two dependent truncated normal random variables. We propose the
use of an easy-to-implement approximation by considering that both variables
are normal and independent having means $\mu_{\beta}$ and $\mu_{\beta-1}$ and
standard deviations $\sigma_{\beta}$ and $\sigma_{\beta-1}$. In this case, the
PDF of the ratio of two normal distributions is given in [21]. Because the
variable $Z$ is non negative, we then use its truncated version given by:

$f_{L}(z)=C(\rho,\mu,\delta_{\beta-1})\frac{\rho}{\pi(1+\rho^{2}z^{2})}exp(-\frac{\rho^{2}\mu^{2}+1}{2\delta_{\beta-1}^{2}})(1+\sqrt{\frac{\pi}{2}}qerf(\frac{q}{\sqrt{2}})exp(\frac{q^{2}}{2}))$
(23)

where $z\geq 0$, $C(\rho,\mu,\delta_{\beta-1})$ a normalization constant,
$q=(1+\mu\rho^{2}z)/\delta_{\beta-1}\sqrt{1+\rho^{2}z^{2}}$,
$\rho=\sigma_{\beta-1}/\sigma_{\beta-1}$, $\mu=\mu_{\beta}/\mu_{\beta-1}$, and
$\delta_{\beta-1}=\sigma_{\beta-1}/\mu_{\beta-1}$.

###  6.2 Observed shape

We propose to define a uncertainty measure of the C-estimator $\theta^{*}$
inspired by Fisher information. For a PDF $h(x|\theta)$, the Fisher
information about $\theta$ is the expected value of second derivative of
$-ln(h(x|\theta))$ with regards to $\theta$, evaluated at $\theta^{*}$. The
interpretation is as follows. The change of the function $ln(h(x|\theta))$ on
the parameter $\theta$ space in the vicinity of $\theta^{*}$ provides
information about the uncertainty of the C-estimator $\theta^{*}$. For
example, the change in orientation of the tangent of $ln(h(x|\theta))$ in the
vicinity of $\theta^{*}$ provides information about the shape of the peak of
$ln(h(x|\ theta))$ at the point $\theta^{*}$. If the change is rapid, then for
any $\theta$ belonging to the vicinity of $\theta^{*}$, $ln(h(x|\theta^{*}))$
can be easily distinguished from any $ln(h(x|\theta))$. In this case, the
accuracy of the estimated $\theta^{*}$, the second derivative of
$ln(h(x|\theta))$ with regards to $\theta$ at the point $\theta^{*}$, and
therefore the Fisher information are all high. The second derivative of
$ln(h(x|\theta))$ with regards to $\theta$ at the point $\theta^{*}$ is the
curvature measure at this point. This geometric interpretation highlights the
strength of Fisher’s information which establishes an equivalence between
uncertainty and this geometric characteristic. It happens that the arithmetic
mean is used instead of the expectation and the Fisher information, known as
observed Fisher information, is the negative of the second derivative of the
logarithm of the likelihood. For the estimation of centrality, we take
inspiration from the observed Fisher information to propose observed C-Fisher
information. Since the uncertainty of the C-estimator $\theta*$ depends on the
curvature of the measured centrality, we define observed C-Fisher information,
as the second derivative of the logarithm of minus the centrality given in Eq.
19. Equivalently, because the centrality is equal to
$lnh(s_{\alpha,\theta}|\theta)$ (See P7 and Q7), then the observed C-Fisher
information, can be seen also as the second derivative of the
$-lnh(s_{\alpha,\theta}|\theta)$.

##  7 Case study

We propose to study the C-estimator of the exponential PDF $h(x|\theta)=\theta
exp(-\theta x)$. This bijective PDF in $x$ was widely used in several areas
including image processing, queuing theory, and physics [3, 1, 2]. The first
derivative of centrality in Eq. 13 is rewritten as follows:

$\frac{\partial C_{\alpha}(\theta)}{\partial\theta}=\left\\{\begin{array}[]{l
r }\sum_{i}\lambda_{i}(-x_{i}+\frac{1}{\theta})(\alpha
g_{\alpha}(x_{i}|\theta)-(\alpha-1)g_{\alpha-1}(x_{i}|\theta))&\mbox{if
$C=L$}\\\
\sum_{i}\lambda_{i}(-x_{i}+\frac{1}{\theta})g_{\alpha}(x_{i}|\theta)&\mbox{if
$C=H$}\end{array}\right.$ (24)

where $g_{\alpha}(x|\theta)=exp(-\alpha\theta
x)/\sum_{i}\lambda_{i}exp(-\alpha\theta x_{i})$. Hence, the critical point is
the following fixed point:

$\theta=\left\\{\begin{array}[]{l r }\frac{1}{\sum_{i}\lambda_{i}x_{i}(\alpha
g_{\alpha}(x_{i}|\theta)-(\alpha-1)g_{\alpha-1}(x_{i}|\theta))}&\mbox{if
$C=L$}\\\ \frac{1}{\sum_{i}\lambda_{i}x_{i}g_{\alpha}(x_{i}|\theta)}&\mbox{if
$C=H$}\end{array}\right.$ (25)

Fig. 1.d depicts a histogram of the absolute value of the DCT coefficients
extracted from a gray-level image. The exponential of the centrality H-C, its
critical points, and its maximums are shown in Figs. 1.a, 1.b, and 1.c as
function of $\alpha$ and $\theta$. Tow observations can be made. First, these
functions are increasing in $\alpha$ as it is expected (P3). Second, some of
critical points are not maximums as it is explicit in Figs. 1.b and 1.c. Fig.
1.e depicts the central observation, denoted $s$ (P7-P9), as function of
$\alpha$ and $\theta$. As expected, a value of $s$ is lower for higher
$\alpha$, because as $\alpha$ increases, the centrality increases (P3) and
hence in the case the centrality $h(s|\theta)$ (P7), which is the exponential
PDF, is higher at a lower $s$. The exponential of L-C in Eq. 8, the critical
points, and the maximums are plotted in Figs. 2.a, 2.b, and 2.c. Fig. 2.h
shows the plots of the critical points $\theta$ in Eq. 25 as functions of
$\alpha$, using the histogram in Fig. 1.d. It can be noted that, the change in
the value of $\theta$ caused by $\alpha$ is faster in the Lehmer case. Fig.
2.h shows that there are estimators that can be derived from L-C but not from
H-C. Moreover, the comparison of the two C-estimators in Eq. 25 leads to
conclude that for $\alpha>1$ (resp. $\alpha<1$) the Lehmer estimator (C=L) is
higher (resp. lower) than that of Hölder (C=H).

a  b  c d e f

Figure 1: First row: The H-C calculated using the histogram (d), critical
points, and maximums. Second row: a histogram, central observations explained
in P7-P9, and H-Fisher at the maximums.

We will now deal with second derivative of the two centralities at the
critical points. Let us set
$v(\theta,\alpha)=\sum_{i}\lambda_{i}x_{i}^{2}g_{\alpha}(x_{i}|\theta)-(\sum_{i}\lambda_{i}x_{i}g_{\alpha}(x_{i}|\theta))^{2}$,
according to Eq . 19, the second derivatives at the critical point are given
by:

$\frac{\partial^{2}C_{\alpha}}{\partial\theta^{2}}=\left\\{\begin{array}[]{l r
}\frac{-2\alpha}{\theta^{2}}+\alpha^{2}\sum_{i}\lambda_{i}x_{i}^{2}g_{\alpha}(x_{i}|\theta)-(\alpha-1)^{2}\sum_{i}\lambda_{i}x_{i}^{2}g_{\alpha-1}(x_{i}|\theta)&\mbox{if
$C=L$}\\\ -\frac{1}{\theta^{2}}+\alpha v(\theta,\alpha)&\mbox{if
$C=H$}\end{array}\right.$ (26)

Note that, $v(\theta,\alpha)>0$. Indeed, let consider the positive function
$\sum_{i}\lambda_{i}g_{\alpha}(x_{i}|\theta)(x_{i}-\sum_{j}\lambda_{j}x_{j}g_{\alpha}(x_{j}|\theta))^{2}$.
It can be written as
$\sum_{i}\lambda_{i}g_{\alpha}(x_{i}|\theta)x_{i}^{2}-2\sum_{i}\lambda_{i}g_{\alpha}(x_{i}|\theta)x_{i}\sum_{j}\lambda_{j}x_{j}g_{\alpha}(x_{j}|\theta)+(\sum_{j}\lambda_{j}x_{j}g_{\alpha}(x_{j}|\theta))^{2}$.
Simplifying it gives
$\sum_{i}\lambda_{i}x_{i}^{2}g_{\alpha}(x_{i}|\theta)-(\sum_{i}\lambda_{i}x_{i}g_{\alpha}(x_{i}|\theta))^{2}=v(\theta,\alpha)$.
When it exists, the critical point of the H-C is a maximum if and only if
$\alpha v(\theta,\alpha)<\frac{1}{\theta^{2}}$. More specifically, there is
always a maximum when $\alpha\leq 0$ or when $\alpha
v(\theta,\alpha)\in[0,\frac{1}{\theta^{2}}[$. Following the same reasoning for
the case of L-C; the critical point is a maximum if
$\sum_{i}\lambda_{i}x_{i}^{2}(\alpha^{2}g_{\alpha}(x_{i}|\theta)-(\alpha-1)^{2}g_{\alpha-1}(x_{i}|\theta))<\frac{2\alpha}{\theta^{2}}$.
From this inequality, we deduce that if the left term is positive then
$\alpha$ is positive and when $\alpha<0$ then the left term is negative. About
the uncertainty, the H-Fisher information estimated at the maximums is plotted
Fig. 1.f as function of $\alpha$ and $\theta$. The uncertainty is lower for
higher $\alpha$. Another important question concerns the comparison of the
C-estimator and the maximum likelihood estimator. We answer this question
through an example in the case of H-Estimator. By using the histogram in Fig.
1.d, the best H-estimator according to the residuals $e_{H}$ in Eq. 20 when
$\beta=2$ is $\theta=0.1300$ obtained at $\alpha=0.0800$ and leading to
$e_{H}=0.0005$. For comparison purposes, the maximum likelihood estimator
(i.e. $\alpha=0$) is $\theta=0.04$, which gives $e_{H}=0.0214$ for $\beta=2$.
The estimated PDFs are plotted in Fig. 1.d.

a b c h

Figure 2: First row: The L-C calculated using the histogram in 1.d., critical
points, and maximums. Second row: the critical points $\theta$ of Hölder
(solid) and Lehmer (dash) centralities as function of $\alpha$ drawn using the
implicit function in Eq. 25.

We end this case study by presenting an example of histogram adjustment of the
absolute values of the DCT coefficients. To do this, we examine the
distribution of the parameter $\alpha$ leading to the best H-estimator
according to $e_{H}()$ in Eq. 20. Fig. 3 presents the best H-estimators of 405
histograms of DCT coefficients extracted from gray-level images. In Fig. 3.a,
an abscissa refers to a histogram and the corresponding ordinate is the value
of the best H-estimator. For a histogram, the maximums of H-C are located and
the one with the smallest $e_{H}()$ is chosen. The average $e_{H}()$ for the
405 histograms is 0.0008. The histogram of the variable $\alpha$ used to
calculate the 405 best H-estimators is shown in Fig. 3.b. It can be observed
that most of the values of $\alpha$ are not zero. Which indicates that,
according to $e_{H}()$, maximum likelihood is not the best estimator.

.a .b

Figure 3: Fitting of 405 histograms of DC coefficients. a) The H-estimator
$\theta$ for each image. b) The histogram of the $\alpha$ selected according
to the residual error.

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