Fuzzy data, prevalent in social sciences and other fields, capture uncertainties arising from subjective evaluations and measurement imprecision. Despite significant advancements in fuzzy statistics, a unified inferential regression-based framework remains undeveloped. Hence, we propose a novel approach for analyzing bounded fuzzy variables within a regression framework. Building on the premise that fuzzy data result from a process analogous to statistical coarsening, we introduce a conditional probabilistic approach that links observed fuzzy statistics (e.g., mode, spread) to the underlying, unobserved statistical model, which depends on external covariates. The inferential problem is addressed using Approximate Bayesian methods, mainly through a Gibbs sampler incorporating a quadratic approximation of the posterior distribution. Simulation studies and applications involving external validations are employed to evaluate the effectiveness of the proposed approach for fuzzy data analysis. By reintegrating fuzzy data analysis into a more traditional statistical framework, this work provides a significant step toward enhancing the interpretability and applicability of fuzzy statistical methods in many applicative contexts.

In experimental practice, especially where the human factor plays an important role, we are faced with the need to analyze phenomena burdened with two types of uncertainty simultaneously: randomness and a lack of precision. While probability theory deals with randomness, and we cope with imprecision using the theory of fuzzy sets, combining both these descriptions is neither straightforward nor simple. Even seemingly simple tasks, such as calculating the probability of an event for a fuzzy random variable, are not obvious. In this contribution, we propose a method of calculating probabilities related to the so-called LR fuzzy random variables. Besides indicating the general method, we show how, in situations where it is difficult to obtain an analytical solution, it is possible to determine the desired probability using numerical methods, referring to the Monte Carlo simulations. The considered approach is illustrated with examples, also based on the real-life dataset.

In this article, we propose two new resampling algorithms for the simulation of bootstrap-like samples of interval-valued fuzzy numbers (IVFNs). These methods (namely, the d-method and the s-method) reuse a primary sample (an initial set) of IVFNs to generate a secondary sample, which also consists of this type of fuzzy numbers, and simultaneously utilize existing dependencies in pairs of some characteristic points of IVFNs. During a corresponding resampling step, a nonparametric approach is used. Additionally, we apply a widely used assumption about the Gaussian kernel densities. The proposed methods in some way resemble Efron's bootstrap, but, contrary to this classical approach, they generate “not exactly the same as previous” IVFNs, so it leads to a greater diversity of the obtained secondary sample. We also numerically check the quality of the introduced methods using a few more statistically oriented approaches together with four similarity measures and three types of IVFNs.

In this paper, we propose two new nonparametric resampling methods for the simulation of bootstrap-like samples of fuzzy numbers. The generated secondary samples are based on an input set (i.e., a primary sample) consisting of left–right fuzzy numbers. The proposed approaches utilize random simulations in a way which, to some extent, resembles a bootstrap. However, contrary to the classical bootstrap approach, the proposed methods are based on alpha-cuts of fuzzy numbers, which are generated in a new nonparametric way. Therefore, these procedures give us an opportunity to create ”not exactly the same as previous” fuzzy numbers and also lead to greater diversity of the obtained output. Moreover, we check whether the introduced methods can be successfully applied in two statistical tests about the mean value of a population of fuzzy numbers.

Catastrophe bonds are financial instruments, which enable to transfer the natural catastrophe risk to financial markets. This paper is a continuation of our earlier research concerning catastrophe bond pricing. We assume the absence of arbitrage and neutral attitude of investors toward catastrophe risk. The interest rate behavior is described by the two-factor Vasicek model. To illustrate and analyze obtained results, we conduct Monte Carlo simulations, using parameters fitted for real data on natural catastrophes. Besides the crisp cat bond pricing formulas, we obtain their fuzzy counterparts, taking into account the uncertainty on the market. Moreover, we propose an automated approach for decision making in fuzzy environment with relevant examples presenting this method.