The robust standard deviations are very similar to the ones obtained using normal distribution theory, indicating that the main result of this analysis is robust to the assumption of multivariate normality.
The method is based on a systematic perturbation expansion around the deterministic, non-spatial mean-field theory, using the theory of distributions to account for space and the underlying stochastic differential equations to account for stochasticity.
Lots of results have been obtained by using value distribution theory (see, e.g., [5] and [6, 7]).
The research on the value distribution of q meromorphic function is very active in the field of complex analysis; many mathematicians had done a lot of works in this project by using the Nevanlinna value distribution theory and obtained many famous results, such as the Picard theorem, Julia direction, Borel theorem, Borel direction, Hayman theorem, Yang-Zhang theorem, and so on (see [3 12]).
The cutoff between low and high expression of a probe is determined from the distribution using the theory of mixture models.
The copula entropy theory can be applied in different ways: (1) the marginal distributions are derived using the entropy theory and the joint distribution using the copula theory (e.g., Hao and Singh 2012; Zhang and Singh 2012).
In general, we can apply the copula entropy theory in three ways: (i) The marginal distributions are derived using the entropy theory, while the joint distribution (i.e., copula function) is modeled through the parametric copula function with its parameter estimated using the Full-MLE, Two-Stage MLE, or Pseudo-MLE.
The discontinuities are treated using the theory of distributions.
The stochastic nature of stream flow discharge is described by the probability density function (PDF) obtained using the theory of distributions.
Halburd and Korhonen [15 17] used value distribution theory and a reasoning related to the singularity confinement to single out the difference Painlevé I and II equations from difference equation (1).
