Stability of random membership frequency and fuzzy statistics (Q1116588)

The aim of the paper is to estimate a membership function of a fuzzy set by statistical methods. Let X be a totally ordered set. A mapping \(\Gamma\) from a probability space into a set of intervals in X is called a random interval if it satisfies a measurability condition. The author shows that, given a random sample from \(\Gamma\), for all \(x\in X\) the random frequency of x converges to P(x\(\in \Gamma)\) with probability one. A [0,1]-valued random variable \(\lambda\) is a \(\Gamma\)-random degree of membership of a fuzzy set A if \(\Gamma\) (\(\omega)\) is the \(\lambda\) (\(\omega)\)-level set of A for all \(\omega\). Using a random sample from \(\Gamma\) and one of \(\lambda\) the author gives a method for estimating a membership function of a holomorphic fuzzy set.