Narrow Results

The goal of this work is the further development of neoclassical analysis, which extends the scope and results of the classical mathematical analysis by applying fuzzy logic to conventional mathematical objects, such as functions, sequences, and series. This allows us to reflect and model vagueness and uncertainty of our knowledge, which results from imprecision of measurement and inaccuracy of computation. Basing on the theory of fuzzy limits, we develop the structure of statistical fuzzy convergence and study its properties. Relations between statistical fuzzy convergence and fuzzy convergence are considered in the First Subsequence Theorem and the First Reduction Theorem. Algebraic structures of statistical fuzzy limits are described in the Linearity Theorem. Topological structures of statistical fuzzy limits are described in the Limit Set Theorem and Limit Fuzzy Set theorems. Relations between statistical convergence, statistical fuzzy convergence, ergodic systems, fuzzy convergence and convergence of statistical characteristics, such as the mean (average), and standard deviation, are studied in Secs. 2 and 4. Introduced constructions and obtained results open new directions for further research that are considered in the Conclusion.

In this study, a new regression method called Kappa regression is introduced to model conditional probabilities. The regression function is based on Dombi’s Kappa function, which is well known in fuzzy theory. Here, we discuss how the Kappa function relates to the Logistic function as well as how it can be used to approximate the Logistic function. We introduce the so-called Generalized Kappa Differential Equation and show that both the Kappa and the Logistic functions can be derived from it. Kappa regression, like binary Logistic regression, models the conditional probability of the event that a dichotomous random variable takes a particular value at a given value of an explanatory variable. This new regression method may be viewed as an alternative to binary Logistic regression, but while in binary Logistic regression the explanatory variable is defined over the entire Euclidean space, in the Kappa regression model the predictor variable is defined over a bounded subset of the Euclidean space. We will also show that asymptotic Kappa regression is Logistic regression. The advantages of this novel method are demonstrated by means of an example, and afterwards some implications are discussed.