The one-way analysis of variance (ANOVA) is mainly based on several assumptions and can be used to compare the means of two or more independent groups of a factor. To relax the normality assumption in one-way ANOVA, recent studies have considered exponential distortion or tilt of a reference distribution. The reason for the exponential distortion was not investigated before; thus the main objective of the study is to closely examine the reason behind it. In doing so, a new generalized semi-parametric approach for one-way ANOVA is introduced. The proposed method not only compares the means but also variances of any type of distributions. Simulation studies show that proposed method has favorable performance than classical ANOVA. The method is demonstrated on meteorological radar data and credit limit data. The asymptotic distribution of the proposed estimator was determined in order to test the hypothesis for equality of one sample multivariate distributions. The power comparison of one sample multivariate distributions reveals that there is a significant power improvement in the proposed chi-square test compared to the Hotelling's T-Square test for non normal distributions. A bootstrap paradigm is incorporated for testing equidistributions of multiple samples. As far as power comparison simulations for multiple large samples are considered, the proposed test outperforms other existing parametric, nonparametric and semi-parametric approaches for non normal distributions.