Bootstrap Kuiper Testing of the Identity of 1D Continuous Distributions using Fuzzy Samples

Abstract

This paper aims to statistically test the null hypothesis for identity of the probability distribution of one-dimensional (1D) continuous parameters in two different populations, presented by fuzzy samples of i.i.d. observations. A degree of membership to the corresponding population is assigned to any of the observations in the fuzzy sample. The test statistic is the Kuiper's statistic, which measures the identity between the two sample cumulative distribution functions (CDF) of the parameter. A Bootstrap algorithm is developed for simulation-based approximation for the CDF of the Kuiper statistic, provided that is true. The of the statistical test is derived using the constructed conditional distribution of the test statistic. The main idea of the proposed Bootstrap test is that, if is true, then the two available fuzzy samples can be merged into a unified fuzzy sample. The latter is summarized into a conditional sample distribution of the 1D continuous parameter used for generation of synthetic pairs of fuzzy samples in different pseudo realities. The proposed algorithm has four modifications, which differ by the method to generate the synthetic fuzzy sample and by the type of the conditional sample distribution derived from the unified fuzzy sample used in the generation process. Initial numerical experiments are presented which tend to claim that the four modifications produce similar results.

Keywords

• fuzzy samples
• percentile Bootstrap procedure
• simulation-based algorithm
• resemblance of fuzzy samples