In the present paper, we would like to draw attention to a possible generalized Fisher information that fits well in the formalism of nonextensive thermostatistics. This generalized Fisher information is defined for densities on Just as the maximum Rényi or Tsallis entropy subject to an elliptic moment constraint is a generalized q-Gaussian, we show that the minimization of the generalized Fisher information also leads a generalized q-Gaussian. This yields a generalized Cramér-Rao inequality. In addition, we show that the generalized Fisher information naturally pops up in a simple inequality that links the generalized entropies, the generalized Fisher information, and an elliptic moment. Finally, we give an extended Stam inequality. In this series of results, the extremal functions are the generalized q-Gaussians. Thus, these results complement the classical characterization of the generalized q-Gaussian and introduce a generalized Fisher information as a new information measure in nonextensive thermostatistics.
REFERENCES
1.
M.
Agueh
, “Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations
,” NoDEA-Nonlinear Diff.
15
(4–5
), 457
–472
(2008
).2.
R. D.
Benguria
and H.
Linde
, “Isoperimetric inequalities for eigenvalues of the Laplace operator
,” in Fourth Summer School in Analysis and Mathematical Physics: Topics in Spectral Theory and Quantum Mechanics
, Contemporary Mathematics
Vol. 476 (American Mathematical Society
, October 2008
), pp. 1
–40
.3.
D. E.
Boekee
, “An extension of the Fisher information measure
,” in Topics in Information Theory
, edited by I
Csiszár
and P.
Elias
(Keszthely
, Hungary
, 1977
), Vol. 16, pp. 113
–123
. János Bolyai Mathematical Society and North-Holland.4.
5.
M.
Casas
, L.
Chimento
, F.
Pennini
, A.
Plastino
, and A. R.
Plastino
, “Fisher information in a Tsallis non-extensive environment
,” Chaos, Solitons Fractals
13
(3
), 451
–459
(March 2002
).6.
L.
Chimento
, F.
Pennini
, and A.
Plastino
, “Naudts-like duality and the extreme Fisher information principle
,” Phys. Rev. E
62
(5 Pt B
), 7462
–7465
(November 2000
).7.
M.
Cohen
, “The Fisher information and convexity
,” IEEE Trans. Inf. Theory
14
(4
), 591
–592
(1968
).8.
D.-P.
Covei
, “Existence and uniqueness of solutions for the Lane, Emden and Fowler type problem
,” Nonlinear Anal. Theory, Methods Appl.
72
(5
), 2684
–2693
(March 2010
).9.
M.
Del Pino
and J.
Dolbeault
, “Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions
,” J. Math. Pures Appl.
81
(9
), 847
–875
(September 2002
).10.
M.
Del Pino
and J.
Dolbeault
, “The optimal euclidean Lp-Sobolev logarithmic inequality
,” J. Funct. Anal.
197
(1
), 151
–161
(January 2003
).11.
A.
Dembo
, T. M.
Cover
, and J. A.
Thomas
, “Information theoretic inequalities
,” IEEE Trans. Inf. Theory
37
(6
), 1501
–1518
(1991
).12.
B. R.
Frieden
, Science from Fisher Information: A Unification
(Cambridge University Press
, 2004
).13.
S.
Furuichi
, “On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics
,” J. Math. Phys.
50
(1
), 013303
–013312
(2009
).14.
S.
Furuichi
, “On generalized Fisher informations and Cramér-Rao type inequalities
,” J. Phys.: Conf. Ser.
201
, 012016
(2010
).15.
I.
Gentil
, “The general optimal Lp-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations
,” J. Funct. Anal.
202
(2
), 591
–599
(2003
).16.
S.
Golomb
, “The information generating function of a probability distribution
,” IEEE Trans. Inf. Theory
12
(1
), 75
–77
(1966
).17.
P.
Hammad
, “Mesure d'ordre α de l'information au sens de Fisher
,” Rev. Stat. Appl.
26
(1
), 73
–84
(1978
).18.
G. H.
Hardy
, J. E.
Littlewood
, and G.
Pólya
, Inequalities
(Cambridge University Press
, February 1988
).19.
P. J.
Huber
and E. M.
Ronchetti
, Robust Statistics
, 2nd ed. (Wiley
, February 2009
).20.
S.
Kesavan
, Symmetrization And Applications
(World Scientific
, August 2006
).21.
C. P.
Kitsos
and N. K.
Tavoularis
, “Logarithmic Sobolev inequalities for information measures
,” IEEE Trans. Inf. Theory
55
(6
), 2554
–2561
(2009
).22.
P.
Lindqvist
, Notes on the p-Laplace equation. Report 102, University of Jyväskylä, 2006
, see http://www.math.ntnu.no/~lqvist/p-laplace.pdf.23.
E.
Lutwak
, D.
Yang
, and G.
Zhang
, “Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information
,” IEEE Trans. Inf. Theory
51
(2
), 473
–478
(2005
).24.
E.
Lutwak
, D.
Yang
, and G.
Zhang
, “Moment-entropy inequalities for a random vector
,” IEEE Trans. Inf. Theory
3
(4
), 1603
–1607
(April 2007
).25.
E.
Lutwak
, S.
Lv
, D.
Yang
, and G.
Zhang
, “Extensions of Fisher information and Stam's inequality
,” IEEE Trans. Inf. Theory
53
(4
), 1319
–1327
(March 2012
).26.
E.
Lutz
, “Anomalous diffusion and Tsallis statistics in an optical lattice
,” Phys. Rev. A
67
(5
), 051402
(2003
).27.
A.
Nachman
and A.
Callegari
, “A nonlinear singular boundary value problem in the theory of pseudoplastic fluids
,” SIAM J. Appl. Math.
38
(2
), 275
–281
(April 1980
).28.
B.
Sz Nagy
, “Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung
,” Acta Sci. Math.
10
, 64
–74
(1941
).29.
J.
Naudts
, “Generalised exponential families and associated entropy functions
,” Entropy
10
(3
), 131
–149
(2008
).30.
J.
Naudts
, “The q-exponential family in statistical physics
,” Cent. Eur. J. Phys.
7
(3
), 405
–413
(2009
).31.
A.
Ohara
and T.
Wada
, “Information geometry of q-Gaussian densities and behaviors of solutions to related diffusion equations
,” J. Phys. A: Math. Theor.
43
(3
), 035002
(2010
).32.
F.
Pennini
, A. R.
Plastino
, and A.
Plastino
, “Rényi entropies and Fisher informations as measures of nonextensivity in a Tsallis setting
,” Physica A
258
(3–4
), 446
–457
(September 1998
).33.
F.
Pennini
, A.
Plastino
, and G. L.
Ferri
, “Semiclassical information from deformed and escort information measures
,” Physica A
383
(2
), 782
–796
(September 2007
).34.
A.
Plastino
, A. R.
Plastino
, and M.
Casas
, “Fisher variational principle and thermodynamics
,” in Variational and Extremum Principles in Macroscopic Systems
, edited by S.
Sieniutycz
and H.
Farkas
(Elsevier
, Oxford
, 2005
), pp. 379
–394
.35.
V.
Schwämmle
, F. D.
Nobre
, and C.
Tsallis
, “q-Gaussians in the porous-medium equation: Stability and time evolution
,” Eur. Phys. J. B: Condens. Matter Complex Sys.
66
(4
), 537
–546
(2008
).36.
A. J.
Stam
, “Some inequalities satisfied by the quantities of information of Fisher and Shannon
,” Inf. Control
2
(2
), 101
–112
(1959
).37.
C.
Tsallis
, “Possible generalization of Boltzmann-Gibbs statistics
,” J. Stat. Phys.
52
(1
), 479
–487
(1988
).38.
39.
C.
Vignat
and A.
Plastino
, “Why is the detection of q-Gaussian behavior such a common occurrence?
,” Physica A
388
(5
), 601
–608
(2009
).© 2012 American Institute of Physics.
2012
American Institute of Physics
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