Gauss’ law of error revisited in the framework of Sharma-Taneja-Mittal information measure

Abstract

Abstract:We reformulate the Gauss' law of error in presence of correlations which are taken into account by means of a deformed product arising in the framework of the Sharma-Taneja-Mittal measure. Having reviewed the main proprieties of the generalized product and its related algebra, we derive, according to the Maximum Likelihood Principle, a family of error distributions with an asymptotic power-law behavior.

Lie symmetries and related group-invariant solutions of a nonlinear Fokker-Planck equation based on the Sharma-Taneja-Mittal entropy

“…In addition, for the Uhlenbeck-Ornstein process, this stationary solution is nothing but a generalized Gaussian which maximizes the corresponding constrained entropic form [26,27]. Remarkably, the NFPE we are introducing is characterized by a nonlinear diffusive term with two different power factors which make this equation more difficult to study as compared to the NFPE for poroses media (known as porous medium equation) which has only a single power factor dependency [25].…”

Law of Multiplicative Error and Its Generalization to the Correlated Observations Represented by the q-Product

“…Tsallis statistics describes a strongly correlated system exhibiting power-law behavior, which results in the q-Gaussian distribution as the distribution of an additive error in q-statistics [12]. Along similar lines, the laws of error for other generalized statistics are also presented in [13,14]. The q-Gaussian distribution provides us not only with a one-parameter generalization of the standard Gaussian distribution (q = 1), but also with a nice unification of important probability distributions such as the Cauchy distribution (q = 2), the t-distribution (q = 1 + 2 n+1…”

“…Similar to above, the ln E (m) i defined by Equation (14) are identical random variables, so every ln E (15) and (16), respectively, the q-likelihood function, L (m) q , for a multiplicative error is defined by…”

Entropic Forms and Related Algebras

“…Recently [29,32], a possible generalization of the central limit theorem has been proposed in order to justify the recurrence of non-Gaussian distributions [39][40][41] in the limit of a large number of statistically dependent events.…”

“…These algebras found several applications running from the number theory [25][26][27], Laplace transformations [28], Fourier transformations [29][30][31], central limit theorem [32,33], combinatorial analysis [34][35][36][37][38], Gauss law of errors [39][40][41]. On the physical ground, generalized algebras have been applied in statistical mechanics [42] and statistical field theory [43].…”