Estimators, escort probabilities, and phi-exponential families in statistical physics

The lower bound of Cramer and Rao is generalized to pairs of families of probability distributions, one of which is escort to the other. This bound is optimal for certain families, called φ-exponential in the paper. Their dual structure is explored. They satisfy a variational principle with respect to an appropriately chosen entropy functional, which is the dual of a free energy functional.

[1]  J. Gibbs.  Elementary Principles in Statistical Mechanics , 1902 .

[2]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[3]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[4]  C. Beck,et al.  Thermodynamics of chaotic systems : an introduction , 1993 .

[5]  M. Murray,et al.  Differential Geometry and Statistics , 1993 .

[6]  C. Tsallis.  What are the Numbers that Experiments Provide , 1994 .

[7]  Giovanni Pistone,et al.  An Infinite-Dimensional Geometric Structure on the Space of all the Probability Measures Equivalent to a Given One , 1995 .

[8]  D. Vernon.  Inform , 1995, Encyclopedia of the UN Sustainable Development Goals.

[9]  C. Tsallis,et al.  The role of constraints within generalized nonextensive statistics , 1998 .

[10]  C. Tsallis.  Generalized entropy-based criterion for consistent testing , 1998 .

[11]  M. Shiino.  H-Theorem with Generalized Relative Entropies and the Tsallis Statistics , 1998 .

[12]  J. Naudts.  Deformed exponentials and logarithms in generalized thermostatistics , 2002, cond-mat/0203489.

[13]  Euclidean and Riemannian Geometrical Approaches to Non-Extensive Thermo-Statistical Mechanics , 2002, cond-mat/0203536.

[14]  J. Naudts.  CONTINUITY OF A CLASS OF ENTROPIES AND RELATIVE ENTROPIES , 2002, math-ph/0208038.

[15]  Sumiyoshi Abe.  Geometry of escort distributions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  A. ADoefaa,et al.  ? ? ? ? f ? ? ? ? ? , 2003 .

[17]  J. Naudts.  Generalized thermostatistics based on deformed exponential and logarithmic functions , 2003, cond-mat/0311438.

[18]  J. Naudts.  Generalized thermostatistics and mean-field theory , 2002, cond-mat/0211444.