Fuzzy Systems and its Applications

Fuzzy Systems and its Applications

A framework for power analysis in fuzzy hypothesis testing through the integration of interval-based tests.

Document Type : Original Article

Authors
Zohre Farahi
Rahim Chinipardaz
Jalal Chachi
Faculty of Mathematical Sciences and Computer, Department of Statistics, Shahid Chamran University of Ahvaz, Ahvaz, Iran
10.22034/jfsa.2026.587866.1306
Abstract
This paper extends the classical equivalence (bioequivalence) test to fuzzy hypotheses. In this framework, the fuzzy null hypothesis is represented in an imprecise manner by means of a fuzzy set. It is then decomposed, using its $\alpha$-cuts, into a collection of precise interval hypotheses. For each $\alpha \in [0, 1]$, a confidence interval at level $(1 - 2\delta)$ is constructed for the parameter of interest; the decomposed interval null hypothesis is then accepted at level $\delta$, provided that the corresponding confidence interval lies within the given $\alpha$-cut interval. The main innovation of this research is the formal definition and analysis of an interval-valued power function for the proposed testing procedure. Although power analysis is fundamental to classical hypothesis testing, its generalization to fuzzy hypotheses has rarely been investigated. Accordingly, an interval-valued power function is introduced that quantifies the probability of correctly accepting the fuzzy null hypothesis across all assumed $\alpha$ levels. This function is obtained by first computing the classical power for each interval test corresponding to a specific $\alpha$-cut, and then combining these results across the range $\alpha \in [0, 1]$, thereby yielding an integrated measure of test performance for the fuzzy hypothesis. Overall, the proposed method provides a coherent bridge between standard interval equivalence testing, statistical inference based on fuzzy information, and test power analysis. The performance of the proposed method is demonstrated in an applied study in soil science, using real data from the Silakhor region (Lorestan, Iran). The results of this example confirm the method's capability to provide fuzzy decision-making with degrees of acceptance/rejection, beyond classical binary decisions.
Keywords
Interval power function
Fuzzy hypotheses
Interval hypotheses
Fuzzy test function
Subjects
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Fuzzy Systems and its Applications
Volume 9, Issue 1 - Serial Number 18
Open Access Statement
June 2026
Pages 115-133

  • Receive Date 20 June 2026
  • Revise Date 24 July 2026
  • Accept Date 03 August 2026