F-tests: A Comprehensive Review of Theory, Applications, and Statistical Inference

The F-test is a fundamental statistical procedure used to compare variances and test hypotheses in various statistical contexts. This paper provides a comprehensive review of F-tests, including their theoretical foundations, mathematical formulations, and practical applications across different domains. We examine the classical F-test for comparing two variances, the F-test in analysis of variance (ANOVA), regression analysis, and model comparison. The paper discusses the assumptions underlying F-tests, their robustness properties, and limitations. Through detailed mathematical exposition and practical examples, we demonstrate the versatility and importance of F-tests in modern statistical analysis. Additionally, we review recent developments and variations of F-tests, including robust alternatives and their applications in contemporary statistical practice.