A Goodness-of-Fit Test Based on Fuzzy Random Variables

Abstract

During the last decades, several methods have been proposed for Kolmogorov−Smirnov one-sample test based on fuzzy random variables to describe the impression of classical random variables. However, such techniques do not discuss the modeling of imprecise observations and simulation of such data from the distribution of a fuzzy random variable. Moreover, such methods rely on a fuzzy cumulative distribution function with known parameters. In this paper, however, a modified Kolmogorov−Smirnov one-sample test is introduced based on a novel notion of fuzzy random variables which comes down to model fuzziness and randomness in the distribution of population in a frequently used family of probability distributions called location and scale distribution functions. A method of moment estimator was also utilized to estimate the location and scale parameters. Then, a notion of non-fuzzy Kolmogorov−Smirnov one-sample test was developed based on fuzzy hypotheses. Monte Carlo simulation was also employed to evaluate the critical value corresponding to a significance level and the performance of the test using power studies. Comparing the observed test statistics and the given fuzzy significance level, a classical procedure was finally used to accept or reject the null fuzzy hypothesis. Two numerical examples including a simulation study and an applied example were provided to clarify the discussions in this paper. The proposed method was also compared with some existing methods. The goodness-of-fit results demonstrated that the proposed Kolmogorov−Smirnov provides an efficient tool to handle statistical inference fuzzy observations.

Keywords

• location and scale
• fuzzy random variable
• fuzzy cumulative distribution function
• fuzzy empirical distribution function
• kolmogorov−smirnov statistics
• critical value
• power of test
• moment estimation