T – CHI – SQUARE FAMILY OF DISTRIBUTIONS USING T – R {Y} FRAMEWORK

Abstract

 In statistical modeling and data analysis, the need for flexible probability distributions that can accurately model complex data structures, particularly in biomedical and survival studies has become increasingly critical. Traditional parametric distributions often fall short in capturing the full variability of data that exhibit characteristics such as bimodality and longitudinal dependencies, especially in medical and survival data. This research addresses this gap by developing a novel convoluted parametric family of continuous probability distributions capable of modeling such complex data structures. The study employs the T-R{Y} framework to construct the new distribution, utilizing the chi-square distribution as the baseline distribution (R), the normal distribution as the transformer (T), and the logistic distribution as the convolution generator (Y). The resulting T-R{Y} distribution is analytically derived and shown to have significant relevance to survival and hazard functions, thereby enhancing its utility in survival analysis. The distribution’s structure enables it to model data with two peaks (bimodality) and account for time-related changes (longitudinal characteristics), which are common in medical datasets. This new distribution was tested and demonstrated to provide better fit and greater flexibility than existing models in handling variations in bimodal, longitudinal survival data. This new distribution offers a promising tool for statisticians and researchers working in medical, biological, and reliability studies.