Dataset Information

Central Limit Theorems under additive deformations.

ABSTRACT: Additive deformations of statistical systems arise in various areas of physics. Classical central limit theory is then no longer applicable, even when standard independence assumptions are preserved. This paper investigates ways in which deformed algebraic operations lead to distinctive central limit theory. We establish some general central limit results that are applicable to a range of examples arising in nonextensive statistical mechanics, including the addition of momenta and velocities via Kaniadakis addition, and Tsallis addition. We also investigate extensions to random additive deformations, and find evidence (based on simulation studies) for a universal limit specific to each statistical system.

SUBMITTER: Eck DJ

PROVIDER: S-EPMC4972458 | biostudies-literature | 2016 Nov

REPOSITORIES: biostudies-literature

ACCESS DATA

Publications

Central Limit Theorems under additive deformations.

Eck Daniel J DJ McKeague Ian W IW

Statistics & probability letters 20161101

Additive deformations of statistical systems arise in various areas of physics. Classical central limit theory is then no longer applicable, even when standard independence assumptions are preserved. This paper investigates ways in which deformed algebraic operations lead to distinctive central limit theory. We establish some general central limit results that are applicable to a range of examples arising in nonextensive statistical mechanics, including the addition of momenta and velocities via ...[more]

PMID: 27499566

Similar Datasets

Project description:In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, ?x. This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull "slope" parameter, ?, can be approximated by ?(Noise) x ?(Transducer), where ?(Noise) is the ? of the Weibull function that fits best to the cumulative noise distribution, and ?(Transducer) depends on the transducer. We derive general expressions for ?(Noise) and ?(Transducer), from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when d' ? (?x)(b), ? ? ?(Noise) x b. We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4-0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull ? reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of ? for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian.

Dataset Information

Central Limit Theorems under additive deformations.

Publications

Central Limit Theorems under additive deformations.

Similar Datasets

OmicsDI is part of the ELIXIR infrastructure

Tweets